# Geometric Evolution Equations with an Internal Dynamic I: $C^{1,1}$ Interpositions for Anisotropic Euclidean Metrics

Théo Belin

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Théo Belin. Geometric Evolution Equations with an Internal Dynamic I: $C^{1,1}$ Interpositions for Anisotropic Euclidean Metrics. 2025. hal-05331783v2

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Geometric Evolution Equations with an Internal Dynamic I: $C^{1,1}$ Interpositions for Anisotropic Euclidean Metrics *

Théo BELIN¹

¹Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Stuttgart, Germany

December 15, 2025

Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>1.1</td><td>Anisotropic distance on $\mathbb{R} \times \mathbb{R}^N$ and non-characteristic $C^{1,1}$ sets</td><td>6</td></tr><tr><td>1.2</td><td>Main results</td><td>9</td></tr><tr><td>1.2.1</td><td>Uniform positivity of $\varepsilon_0$ and density of $\Sigma_\varepsilon$</td><td>10</td></tr><tr><td>1.2.2</td><td>Independence of interpositions away from critical times</td><td>11</td></tr><tr><td>2</td><td>Preliminary results</td><td>12</td></tr><tr><td>2.1</td><td>Density of distances from A to the boundary of $C^{1,1}$ interpositions</td><td>12</td></tr><tr><td>2.2</td><td>Junction of partial interpositions</td><td>14</td></tr><tr><td>3</td><td>Proof of Theorem 1.5</td><td>15</td></tr><tr><td>3.1</td><td>Proof of Lemma 3.3</td><td>22</td></tr><tr><td>4</td><td>Proof of Theorem 1.6</td><td>27</td></tr></table>

<footnote>*AMS classification: 53A07, 53E99</footnote>
# Abstract

An interposition lemma is shown for families of euclidean anisotropic distances near singularity in a particular direction. Specifically, between two disjoint closed sets A, B ⊂ ℝ_t × ℝ_x^N, where A is compact, as long as the projections of A and B on ℝ_t have a non-empty intersection, one can find a discontinuous, piecewise C^1,1 and e_t-non-characteristic interposition K_ε between A and B for the anisotropic euclidean metric d^ε((t, x), (s, y)) := (1/ε^2(t - s)^2 + ||x - y||_N^2)^{1/2} for all ε > 0 small enough. Parts of these interpositions can be replaced by cylindrical domains uniformly with respect to ε > 0. In a subsequent work, the interposition result and its variants play a central role in the proof of inclusion principles for viscosity solutions of abstract geometric evolution problems governed by an internal dynamic. The non-local nature of these evolutions in both space and time is the motivation for providing the tools presented here.

# 1 Introduction

Let E = ℝ^N be a euclidean space, N ∈ N, equipped with an anisotropic scalar product ⟨·, ·⟩_N := ⟨A·, ·⟩_N for some self-adjoint and positive definite operator A ∈ L(E), where ⟨·, ·⟩_N is the canonical inner product of ℝ^D. We denote the associated norm as ||·||_N := ⟨·, ·⟩_N^{1/2}, the distance to a set E ⊂ E as d_N^A(·, E) := inf_{p∈E} ||·-p||_N^A and the minimal distance between any two sets E and F as

$$ \mathrm{dist}_{N}^{A}(E, F) := \inf_{(p,q) \in E \times F} \|p - q\|_{N}^{A}, $$

where conventionally we set dist_N^A(E, F) = ∞ if either E or F are empty. We shall also denote for p ∈ E, r ≥ 0, the open A-ball centered at p as

$$ \mathbb{B}_{N}^{A}(p, r) := \{q \in E \mid \|p - q\|_{N}^{A} < r\}, $$

as well as the tubular A-neighborhood of the set E as

$$ E^{A,r} := E + \mathbb{B}_{N}^{A}(p, r). $$

In the canonical case where A = Id, we shall drop the A superscript in all the notations above. Moreover, when the dimension is clear from the context, we shall drop the N subscript.

Finally, E^c denotes the complement of E and we shall use Cl(E) (resp. Int(E)) to denote the closure (resp. interior) of E in ℝ^N. We shall also use E ◦ (resp. E) for a shorter notation.
An A-interposition between two sets $A, B \subset \mathbb{E} = \mathbb{R}^N$ is a third set $K \subset \mathbb{E}$ that satisfies the following conditions:

$$A \subset K \subset B^c \qquad (1.1)$$

and

$$\mathrm{dist}^A(A, B) = \mathrm{dist}^A(A, \partial K) + \mathrm{dist}^A(\partial K, B), \qquad (1.2)$$

where $E \subset F$ means $\mathrm{Cl}(E) \subset \mathrm{Int}_N(F)$ and $\partial K := \mathrm{Cl}(K) \setminus \mathrm{Int}_N(K)$. Since the inequality $\mathrm{dist}^A(A, B) \ge \mathrm{dist}^A(A, \partial K) + \mathrm{dist}^A(\partial K, B)$ holds generally as a consequence of (1.1) and the equality case of the triangle inequality for euclidean metrics, the opposite inequality in (1.2) is actually the important one. Morally speaking (1.2) expresses that the boundary $\partial K$ cannot lie simultaneously close to A and B. Assuming $\mathrm{dist}^A(A, B) > 0$, the condition (1.2) is equivalent to the existence of a $\sigma \in (0, \mathrm{dist}^A(A, B))$ such that for all $p \in \partial K$,

$$d^A(p, A) \ge \sigma \quad \text{and} \quad d^A(p, B) \ge \mathrm{dist}^A(A, B) - \sigma.$$

In particular we see that any $\sigma$-sublevel-sets of $d^A(\cdot, A)$ with $\sigma \in (0, \mathrm{dist}^A(A, B))$ is an A-interpositions between A and B.

Interpositions convey geometric and differential informations locally around critical points of the distance $\mathrm{dist}^A(A, B)$, i.e. around points $p \in \partial A$, $q \in \partial B$ such that $\mathrm{dist}^A(A, B) = \|p - q\|^A$. The surface $\partial K$, when regular, is tangent to $\partial A$ (resp. $\partial B$) at $p$ (resp. at $q$) upon a suitable translation. It thus becomes possible to compare, via $\partial K$, the local differential properties of $\partial A$ and $\partial B$ at these critical points. Unfortunately, the $\sigma$-sublevel-set of $d^A(\cdot, A)$ described above does not generally enjoy better regularity than Lipschitz continuity, and this is insufficient in most applications to obtain interesting information. Ilmanen [15] showed however that under a compactness assumption on A, it is always possible to find an interposition $K$ whose boundary is a $C^{1,1}$ hypersurface.

**Theorem 1.1** (Ilmanen's interposition [15]). Let A and B be two disjoint closed subsets of $\mathbb{R}^N$ with A compact, then there exists a closed set $K$, with a $C^{1,1}$ boundary, which is an interposition between A and B.

The result still holds for any anisotropic euclidean metric $d^A$ by noting that $\mathrm{dist}^A(E, F) = \mathrm{dist}(R(E), R(F))$ where $R \in \mathcal{L}(\mathbb{E})$ is any square root of A, i.e. such that $A = R^T R$.

Note that the $C^{1,1}$ regularity is actually sharp as can be seen on the example in $\mathbb{R}^2$ developed in Figure 1. Theorem 1.1 is an important development for the theory of weak viscosity solutions for geometric evolution equations in both local and spatially non-local settings. Ilmanen in [15] used it for the study of the local mean curvature flow through level-set methods, and was also a crucial tool to show inclusion principles for other geometric evolution equations of first order, meaning here that the domain of classical validity of the equation is not
more regular than $C^{1,1}$ regularity, with local and non-local terms in space. For instance in evolutions of Hele-Shaw type as can be seen in works of Cardaliaguet and coauthors [5, 6, 9, 7, 8]. The result has also had a significant impact in the study of non-local and generalized mean curvature flows as exemplified with the work of Chambolle and coauthors [11, 12, 10]. The proof of Theorem 1.3 can be found in [6] as well as in [2]. It can be seen as a geometric counterpart of a known $C^{1,1}$ interposition result for semi-concave functions (see e.g. [3, 14, 4]).

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>Figure 1: The only interpositions between $A = [-1, 1]^2$ and $B = (A^1)^c$ are the $K_\sigma := A^\sigma$ for $\sigma \in (0, 1)$. Remark that $\partial K_\sigma$ is only $C^{1,1}$ because it is a union of straight lines and circular arcs.</figcaption></figure>

The lemma has been generalized in several directions, in particular in [7, Sec. 3 Theorem 2] where a spatial interposition between two time-dependent sets $A, B \subset \mathbb{R}_t \times \mathbb{R}_x^N$ can be extended locally in the time direction near critical points of $\text{dist}_{N+1}^A(A, B)$.

**Geometric evolutions with an internal dynamic** The generalization Theorem 1.3 and the subsequent improvements (Theorem 1.5, Theorem 1.6) that we will present here, will have important implications in a future work, for the analysis of what we call geometric evolutions with an internal dynamic, i.e. geometric evolution equations of the form

$$V_{\Omega}(t, x) = H(t, x, \Omega, \lambda), \quad x \in \partial[\Omega(t)], \qquad (1.3)$$
where the outward normal velocity $V_\Omega$ of the time-dependent map $t \mapsto \Omega(t)$ at $x \in \partial[\Omega(t)]$ is prescribed by a non-local functional $H$ which is invariant with respect to some internal dynamic, i.e. such that

$$H(t, x, \Omega, \lambda) = H(t - \tau, x, \Omega(\cdot + \tau), \Phi_{\tau,0}^{\Omega}[\lambda]),$$

where $\Phi_{t,t_0}^\Omega: \Lambda \to \Lambda$ is a non-autonomous dynamic, so-called internal because it is $\Omega$ dependent. This model is non-local in space and in time, meaning $H$ depends on spatially non-local features of $\Omega$ (among others volume, perimeter etc.) as well as the past evolution of the set by means of the internal dynamic $\Phi$. A typical application of (1.3) is the one-phase Stefan problem for which $H_S(t, x, \Omega, \lambda) := -\nabla_x u(t, x) \cdot n(t, x)$, where $n \in \mathbb{S}^{N-1}$ is the outward unit normal to $\Omega(t)$ at $x$ and $u$ solves the heat equation with initial condition $\lambda: \mathbb{R}^N \to \mathbb{R}_+$ and homogeneous Dirichlet boundary conditions on the sides of the time-varying domain

$$\bigcup_{t \ge 0} \{t\} \times \Omega(t).$$

The trace of $\nabla_x u$ has sufficient regularity up to the boundary of $\mathcal{T}$ whenever $\mathcal{T}$ has a $C^{1,1}$ boundary with finite velocity (i.e. without characteristic points in the direction of time), thus the equation (1.3) has a classical meaning for this regularity. Viscosity tube solutions to (1.3), in the spirit of [5, 6], can be defined where test tubes have $C^{1,1}$ regularity. Using the terminology developed in [11], such equations are referred to as first-order equations since the $C^2$ regularity is not required.

The viscosity solutions developed will tackle a quite general class of equations, allowing for many variations of the Stefan problem. We should mention in particular that more specific viscosity solutions, for the Stefan problem and other non-local evolution equations, already exist in the literature (see e.g. the work of Kim and coauthors in [16, 17, 18]). Let us further mention that this work is a continuation of a program that was initially explored in [1, Chap. 3].

Remark crucially that the original Theorem 1.1 is not suitable for the construction of non-characteristic or finite-velocity $C^{1,1}$ test sets, because the set of points of infinite velocity on the boundary $\partial\mathcal{K}$ could be very hard to handle. Observe also that the extension provided in [7] is unsuited since it only produces a locally time-extended interposition and cannot adapt to our fully non-local setting.

This is why we must find a way to construct a full space-time interposition with $C^{1,1}$ regularity and with finite velocity, at least around the critical points. The results we prove here are will be central in the proof of an inclusion principle for viscosity solutions of (1.3). We believe however that the results presented here could be of independent use for other problems. For this reason and the length of the proofs, we have decided to separate the results here from the main
applications they were intended for.

**Objective** We will prove a generalization of Theorem 1.1 for a family of anisotropic norms $||\cdot||^\epsilon$ on $\mathbb{R}^{N+1}$ which get arbitrarily close to a singular non-euclidean metric $d^0$. The $C^{1,1}$ interposition is required to be non-characteristic in the direction of anisotropy of the family of distances, here chosen without loss of generality as the first component.

**Outline** In the following subsections in this introduction, we introduce in subsection 1.1 the family of penalized anisotropic distances and the notions of non-characteristic $C^{1,1}$ sets. Then in subsection 1.2 we present our main result Theorem 1.3 along with two improvements Theorem 1.5, Theorem 1.6 relevant for applications. In section 2 we discuss some preliminary results related to interpositions. The section 3 and section 4 are then dedicated to the proofs of the main improvements (Theorem 1.3 being obtained as a corollary to the first of these improvements).

## 1.1 Anisotropic distance on $\mathbb{R} \times \mathbb{R}^N$ and non-characteristic $C^{1,1}$ sets

We look at the euclidean space $\mathbb{E} = \mathbb{R} \times \mathbb{R}^N$, where the first direction $e_t := (1, 0, ..., 0)$ is the direction of anisotropy. For $\epsilon > 0$, the associated scalar product is denoted

$$ \langle (t, x), (s, y) \rangle^{\epsilon} := \frac{ts}{\epsilon^2} + \langle x, y \rangle $$

and is associated with the diagonal operator $A_\epsilon := \text{diag}(\frac{1}{\epsilon^2}, 1, ..., 1)$. For convenience we shall simply denote $\epsilon$ in place of $A_\epsilon$ in our existing notations. In this anisotropic space, any ball $B_{N+1}^\epsilon(p, r)$ is thus flattened in the $t$-direction compared to the usual euclidean ball $B_{N+1}(p, r)$. We also define the metric $d^0$ on $\mathbb{E}$ obtained in the (formal) limit as $\epsilon \to 0$, which is no longer euclidean, by

$$ d^0((t, x), (s, y)) := \begin{cases} +\infty & \text{if } t \neq s, \\ \|x - y\|_N & \text{otherwise,} \end{cases} $$

as well as the associated distance functions $d^0(\cdot, E)$ and $\text{dist}^0(E, F)$ with obvious definitions.

Before stating the main interposition result, we describe the notion of $C^{1,1}$ regularity with finite velocity for sets of $\mathbb{R} \times \mathbb{R}^N$ and an important relaxed piecewise $C^{1,1}$ regularity.

**$C^{1,1}$ regularity with finite velocity** For $E \subset \mathbb{R} \times \mathbb{R}^N$, $t \in \mathbb{R}$ and $I \subset \mathbb{R}$, the $t$-section of $E$ is

$$ E(t) := \{x \in \mathbb{R}^N \mid (t, x) \in E\} \subset \mathbb{R}^N, $$
and the I-restriction of E is

$$E|_{I} := E \cap (I \times \mathbb{R}^{N}).$$

Moreover, we will say that E is locally bounded in t, if $E|_{I}$ is bounded in $\mathbb{R} \times \mathbb{R}^{N}$ for any bounded $I \subset \mathbb{R}$.

**Definition 1.1.** Let $I \subset \mathbb{R}$. A set $\mathcal{K} \subset \mathbb{R} \times \mathbb{R}^{N}$ is $C^{1,1}$ with finite velocity on $I$, or $\mathcal{K} \in C_{t}^{1,1}(I)$ for short, if $\mathcal{K}$ is locally bounded in $t$ and there exists a spatially non-degenerate level-set representation of $\mathcal{K}$, i.e. a map $\psi: I \times \mathbb{R}^{N} \to \mathbb{R}$ such that

(i) For each $p \in \partial_{I}\mathcal{K}$, there is a neighborhood $V_{p}$ of $p$ relative to $I \times \mathbb{R}^{N}$ such that $\psi \in C^{1,1}(V_{p})$,

(ii) there hold the level-set representations

$$\mathrm{Int}_{I}\mathcal{K} = \{p \in I \times \mathbb{R}^{N} \mid \psi(p) < 0\},$$

$$\partial_{I}\mathcal{K} = \{p \in I \times \mathbb{R}^{N} \mid \psi(p) = 0\}$$

(iii) and for all $p \in \partial_{I}\mathcal{K}$,

$$\nabla_{x}\psi(p) \neq 0,$$

where $\mathrm{Int}_{I}\mathcal{K}$, (resp. $\partial_{N+1,I}\mathcal{K} = \mathrm{Cl}_{I} \setminus \mathrm{Int}_{I}$) is the interior (resp. the boundary) of $\mathcal{K}$ relative to the induced topology on $I \times \mathbb{R}^{N}$.

**Remark 1.1.** Let $\mathcal{K} \in C_{t}^{1,1}(I)$ with a spatially non-degenerate level-set representation $\psi$. For any $p_{0} \in \partial_{I}\mathcal{K}$, we can define the unit outward normal vector $\nu(p_{0}) := (\nu_{t}, \nu_{x}) \in \mathbb{S}^{N}$ of $\mathcal{K}$ as

$$\begin{pmatrix} \nu_{t} \\ \nu_{x} \end{pmatrix} = \frac{1}{|D\psi(p_{0})|} \begin{pmatrix} \partial_{t}\psi(p_{0}) \\ \nabla_{x}\psi(p_{0}) \end{pmatrix}$$

where $D = (\partial_{t}, \nabla_{x})^{T}$ is the full gradient. The equation makes sense since $|D\psi(p_{0})| \neq 0$ holds from $\nabla_{x}\psi(p_{0}) \neq 0$. Remark also that $\nu_{x} \neq 0$, thus points of $\partial\mathcal{K}$ are non-characteristic in the direction of time. Finally, the velocity of $p_{0}$ being $-\frac{\nu_{t}}{|\nu_{x}|}$, this justifies the terminology of finite velocity.

It is straightforward to see that if $J \subset I$, then $C_{t}^{1,1}(I)$ injects canonically in $C_{t}^{1,1}(J)$ by means of the restriction operator. Note however that $C_{t}^{1,1}(\bar{I}) \subsetneq C_{t}^{1,1}(I)$ in general, even for intervals, as one can see with the example

$$\mathcal{K}: t \in \mathbb{R}_{+} \mapsto \mathbb{B}_{N}\left(0, 1 + \sqrt{t}\right).$$

We see that $\mathcal{K} \in C_{t}^{1,1}((0, +\infty))$ but $\mathcal{K} \notin C_{t}^{1,1}([0, +\infty))$, because $\partial\mathcal{K}$ has points of infinite velocity at $t = 0$, which are exactly the points $\partial[\mathcal{K}(0)] = \partial\mathbb{B}_{N}(0, 1)$.
Remark also that if $\mathcal{K} \in C_t^{1,1}(\mathbb{R})$, then $\partial_{N+1}\mathcal{K}$ is a manifold of class $C^{1,1}$ in $\mathbb{R} \times \mathbb{R}^N$ and for each $t \in \mathbb{R}$, $\partial[\mathcal{K}(t)]$ is a $C^{1,1}$ manifold in $\mathbb{R}^N$. These facts can be derived from [13, Chap. 2, Theorem 4.2] which is a consequence of the inverse mapping theorem.

**Piecewise $C^{1,1}$ regularity with finite velocity** In general, non-characteristic interpositions between sets of $\mathbb{R} \times \mathbb{R}^N$ cannot be found in $C_t^{1,1}(\mathbb{R})$. A motivation for this fact is the fact that every $t$-section of a $\mathcal{K} \in C_t^{1,1}(\mathbb{R})$ is homeomorphic to every other. Delfour's and Zolésio's [13, Theorem 4.1] actually provide the following proposition.

**Proposition 1.2.** For any $\mathcal{K} \subset \mathbb{R} \times \mathbb{R}^N$ with $\mathcal{K} \in C_t^{1,1}(\mathbb{R})$, there exists a map $X : \mathbb{R} \times \mathbb{R}^N$ such that for each $t \in \mathbb{R}$,

$$X(t, \cdot) : \mathbb{R}^N \mapsto \mathbb{R}^N$$

is a bi-Lipschitz homeomorphism of $\mathbb{R}^N$ with

$$X(t, \mathrm{Cl}(\mathcal{K}(0))) = \mathrm{Cl}(\mathcal{K}(t)).$$

It is unreasonable to expect homeomorphic $t$-sections for interpositions between two sets $A, B \subset \mathbb{R} \times \mathbb{R}^N$ generically. Take for instance $A := \mathbb{B}_{N+1}((0,0), 1) \cup \mathbb{B}_{N+1}((5,0), 1)$, two vertically aligned unit balls and $B := (\mathbb{B}_{N+1}((0,0), 2) \cup \mathbb{B}_{N+1}((5,0), 2))^c$, we can see that any $\varepsilon$-interposition $\mathcal{K}$ between $A$ and $B$ must satisfy

$$\mathcal{K}(t) = \emptyset, \quad \text{for } t \in [2, 3],$$

and thus cannot possibly have homeomorphic $t$-sections, unless it is empty. The relaxation we choose is that of piecewise $C^{1,1}$ sets with finite velocity, which allow for a locally finite number of topological breaking points in time.

**Definition 1.2.** Let $I \subset \mathbb{R}$ be an interval. A set $\mathcal{K} \subset \mathbb{R} \times \mathbb{R}^N$ is said to be piecewise $C^{1,1}$ with finite velocity on $I$, or $\mathcal{K} \in C_{t,pw}^{1,1}(I)$ for short, if for any $J \subset I$, there exists $N_J \ge 1$ and $\min J = t_0 < t_1 < \cdots < t_N = \max J$ such that for each $i \in \{0, \dots, N_J - 1\}$

$$\mathcal{K} \in C_t^{1,1}([t_i, t_{i+1}]).$$

When $N_J$ is minimal, the collection $(t_i)_{0 \le i \le N_J}$ is unique and is called the times of discontinuity of $\mathcal{K}$ on $J$.

Note that for a compact interval $I \subset \mathbb{R}$, $\mathcal{K} \in C_{t,pw}^{1,1}(I)$ implies $\mathcal{K} \in C_t^{1,1}(\bigcup_{i=0}^{N_J}(t_i, t_{i+1}))$, where $(t_i)_i$ are the times of discontinuity of $\mathcal{K}$ on $I$. The converse is however not true because $C_t^{1,1}([t_i, t_{i+1}]) \subsetneq C_t^{1,1}((t_i, t_{i+1}))$.

The introduction of topological breaking points induces the existence of points of infinite velocity on the boundary of $\mathcal{K}$ at the times of discontinuity. We thus need to reinforce the notion of $\varepsilon$-interposition to ensure that the points of tangency of $\mathcal{K}$ have finite velocity. Let

$$\mathcal{D}^{\varepsilon}(A, B) := \{p \in \mathbb{E} \mid d^{\varepsilon}(p, A) + d^{\varepsilon}(p, B) = \mathrm{dist}^{\varepsilon}(A, B)\},$$
be the set of critical points between A and B. Due to the triangle inequality, these are exactly the points of global minimum of $p \mapsto d^\Lambda(p, A) + d^\Lambda(p, B)$. We now define the notion of non-characteristic interposition.

**Definition 1.3.** A set $\mathcal{K} \in \mathbb{R} \times \mathbb{R}^N$ is said to be a non-characteristic $\varepsilon$-interposition between A and B, if

(i) $\mathcal{K}$ is a $\varepsilon$-interposition between A and B,

(ii) for any $p \in \partial\mathcal{K} \cap \mathcal{D}^\varepsilon(A, B)$, the boundary $\partial\mathcal{K}$ is $C^1$ in a neighborhood of $p$ and $\nu_x(p) \neq 0$.

## 1.2 Main results

The main interposition result is now stated as follows.

**Theorem 1.3.** Let $A, B \subset \mathbb{R} \times \mathbb{R}^N$ be two closed and disjoint sets with A compact and such that

$$ \mathrm{dist}^0(A, B) < +\infty, \qquad (1.4) $$

then there exists $\varepsilon_0 > 0$ such that for any $\varepsilon \in (0, \varepsilon_0)$, there exists $\mathcal{K}_\varepsilon \in C_{t,pw}^{1,1}(\mathbb{R})$ which is a non-characteristic $\varepsilon$-interposition between A and B.

The finiteness condition on $\mathrm{dist}^0$ (1.4), ensures the non-existence of vertically aligned critical points for the distance $\mathrm{dist}^\varepsilon(A, B)$ for $\varepsilon > 0$ small enough. This property, as we will see in the next section, is crucial to the existence of non-characteristic $\varepsilon$-interpositions. This condition could likely be relaxed further but it is non-restrictive for the applications in the scope of this work. Additionally, the finiteness condition will have an important role for improved versions of Theorem 1.3.

For applications in geometric evolution problems with internal dynamics, Theorem 1.3 alone is however not sufficient. To improve the result and make it suitable for applications, it will in fact be important to have the following requirements, here presented in a rather informal manner.

R1) The range parameter $\varepsilon_0 > 0$ should uniformly positive on collections of good subsets of A and B.

R2) The set $\Sigma_\varepsilon$ of reals $\sigma$ such that an $\varepsilon$-interposition $\mathcal{K} \in C_{t,pw}^{1,1}(\mathbb{R})$ exists between A and B with $\sigma = \mathrm{dist}^\varepsilon(A, \partial\mathcal{K})$, should be dense in $(0, \mathrm{dist}^\varepsilon(A, B))$.

R3) It should be possible to construct interpositions whose boundary is independent of $\varepsilon > 0$ away from the points in $\mathcal{D}^\varepsilon(A, B) \cap \partial\mathcal{K}$.

In the two following paragraphs, first Theorem 1.5 will provide an answer to R1) and R2) and as such is an improvement of Theorem 1.3. The theorem will be proved in section 3. Note that Theorem 1.3 is presented here (without this improvement) for clarity of the presentation, and will only be proved through
Theorem 1.5. On the other hand, Theorem 1.6 will provide a partial answer to R3) where, under some additional assumptions, the uniform shape can be taken as a cylinder away from critical times, and will be proved in section 4. Before we prove the theorems, in section 2, we provide some basic tools to manipulate interpositions. In a first subsection we give the main arguments giving the density of $\Sigma_{\epsilon}$ and in the second subsection we give some tools for combining two interpositions with the same distance to A.

### 1.2.1 Uniform positivity of $\epsilon_0$ and density of $\Sigma_{\epsilon}$

We make the following observation: if for some $\epsilon > 0$, a non-characteristic $\epsilon$-interposition $\mathcal{K}_{\epsilon} \in C_{t,pw}^{1,1}(\mathbb{R})$ does exist between A and B, then by virtue of the non-characteristic condition stated in Definition 1.3, any couple of points $(p_A = (t_A, x_A), p_B = (t_B, x_B))$ realizing the distance $\text{dist}^{\epsilon}(A, B)$ cannot be vertically aligned (i.e. $x_A \neq x_B$). Let us then record, for $A, B \in \mathbb{R} \times \mathbb{R}^N$, the set of critical points of the distance $\text{dist}^{\epsilon}(A, B)$ in

$$ \Gamma_{\epsilon}(A, B) := \{(p, q) \in \overline{A} \times \overline{B} \mid \|p - q\|^{\epsilon} = \text{dist}^{\epsilon}(A, B)\}, $$

and we will say that $\Gamma_{\epsilon}(A, B)$ is non-characteristic if it does not contain pairs of vertically aligned points, i.e. pairs of the form $((t, x), (s, y))$, with $t, s \in \mathbb{R}$ and $x = y \in \mathbb{R}^N$.

Let us now denote the following threshold,

$$ \epsilon_{\Gamma}(A, B) := \sup\{\epsilon_0 > 0 \mid \forall \epsilon \in (0, \epsilon_0], \Gamma_{\epsilon}(A, B) \text{ is non-characteristic}\}, \quad (1.5) $$

which gives the range of $\epsilon$ near 0 which produce a non-characteristic set $\Gamma_{\epsilon}(A, B)$. The map $\epsilon_{\Gamma}: \mathcal{P}(\mathbb{E}) \times \mathcal{P}(\mathbb{E}) \to \mathbb{R}_+$ is uniformly positive with respect to gradual subsets of given A and B, i.e. subsets $A' \subset A, B' \subset B$ for which $\text{dist}^0(A', B')$ is not too large. To this end we define the M-gradual subsets of (A, B) as follows

$$ \text{Sub}_M(A, B) := \left\{ (A', B') \in \mathcal{P}(\mathbb{E}) \times \mathcal{P}(\mathbb{E}) \mid \begin{array}{l} A' \subset A \text{ and } B' \subset B, \\ \text{dist}^0(A', B') \le M \end{array} \right\}. $$

Remark that, since $d^0$ is non-increasing with respect to set inclusion, $\text{Sub}_M(A, B) \neq \emptyset$ if and only if $M \ge \text{dist}^0(A, B)$. This observation and its proof is summarized in the following lemma.

**Lemma 1.4.** Let $A, B \subset \mathbb{E}$ be closed and disjoint with A compact and let $M \ge \text{dist}^0(A, B)$. There exists $\delta_M := \delta_M(A, B) > 0$ such that

$$ \epsilon_{\Gamma}(A', B') \ge \delta_M, $$

for all $(A', B') \in \text{Sub}_M(A, B)$.

*Proof.* Assume by contradiction that for each $n \in \mathbb{N}$, there exists $0 < \epsilon_n \le \frac{1}{n}$, a pair of subsets $(A_n, B_n) \in \text{Sub}_M(A, B)$ and a pair of vertically aligned points
$(p_n = (t_n, x_n) \in A_n, q_n = (s_n, x_n) \in B_n)$ which realize $\text{dist}^{\epsilon_n}(A_n, B_n)$.

By compactness of A, upon extracting a subsequence, there exists $(t, x) \in A$ such that

$$p_n \to (t, x).$$

Also, since $\text{dist}^{\epsilon_n}(A_n, B_n) \le \text{dist}^0[A_n, B_n] \le M$, we find that for all $n$,

$$ \begin{aligned} |t_n - s_n|^2 &= \epsilon_n^2 \text{dist}^{\epsilon_n}(A_n, B_n)^2 \\ &\le \frac{1}{n^2} M^2. \end{aligned} $$

Hence passing to the limit, we infer that, like $(t_n)_{n \in \mathbb{N}}$, $(s_n)_{n \in \mathbb{N}}$ must converge to $t$. As a result $(q_n)_{n \in \mathbb{N}} \subset B$ must also converge to $(t, x)$. Since $B$ is closed, the contradiction $(t, x) \in A \cap B$ then falls. □

It turns out that the threshold $\epsilon_0$ given in Theorem 1.3 can actually be given by this new uniformly positive threshold $\epsilon_\Gamma$. Also the set of distances between A and $\partial K$ can be taken dense in $(0, \text{dist}^\epsilon(A, B))$.

**Theorem 1.5.** Let $A, B \subset \mathbb{R} \times \mathbb{R}^N$ be two closed disjoint sets with A compact and such that

$$\text{dist}^0(A, B) < +\infty,$$

then for any $\epsilon \in (0, \epsilon_\Gamma(A, B))$, there exists a dense set $\Sigma_\epsilon \subset (0, \text{dist}^\epsilon(A, B))$ such that for each $\sigma \in \Sigma_\epsilon$ there exists $K_{\epsilon, \sigma} \in C_{t,pw}^{1,1}(\mathbb{R})$ which is a non-characteristic $\epsilon$-interposition between A and B and

$$\text{dist}^\epsilon(\partial K_{\epsilon, \sigma}, A) = \sigma.$$

### 1.2.2 Independence of interpositions away from critical times

Let $A, B \subset \mathbb{R} \times \mathbb{R}^N$ be closed and disjoint with A compact. It generally holds that $\text{dist}(A(0), B(0)) \ge \text{dist}^0(A, B)$, but when the strict inequality

$$\text{dist}^0(A, B) < \text{dist}(A(0), B(0))$$

also holds, i.e. when $t = 0$ is not a critical time for $\text{dist}^0(A, B)$, then for $\epsilon > 0$ small enough, the critical points of $\text{dist}^\epsilon(A, B)$ lie away from the hyperplane $\{t = 0\}$. Some room is thus available around $t = 0$ to alter the $\epsilon$-interpositions between A and B, and this can be done in a uniform manner with respect to $\epsilon$. To be precise, denote for $\kappa > 0$ and $E \subset \mathbb{R} \times \mathbb{R}^N$,

$$E_\kappa := E|_{[-\kappa, \kappa]}.$$

In Theorem 1.6 below, the alteration is chosen as a cylinder.
**Theorem 1.6.** Let A, B ⊂ ℝ × ℝ$^N$ be two closed and disjoint sets with A compact and assume that there exists $κ > 0$ such that

$$ \mathrm{dist}^0(A, B) < \mathrm{dist}^0(A_κ, B_κ) < \mathrm{dist}(A(0), B(0)) < +\infty, $$

then there exists $δ = δ[A_κ, B_κ] > 0$, a subset $ω = ω[A_κ, B_κ] ⊂ ℝ^N$ with $C^{1,1}$ boundary and $ε_*= ε_*[A_κ, B_κ] > 0$, all of which only depend on $A_κ$ and $B_κ$ and such that for all $ε ∈ (0, \min\{ε_Γ(A, B), ε_*\})$, there exists a non-characteristic $ε$-interposition $K_ε$ between A and B such that,

$$ K_ε(t) = ω, \quad ∀t ∈ [-δ, δ]. $$

# 2 Preliminary results

In this section we give some preliminary results on interpositions. In subsection 2.1 an improved version of the original result by Ilmanen Theorem 1.1 is given, where it is shown that the set of distances between A and the boundary $∂K$ of the interposition can be chosen in a dense subset of $(0, \mathrm{dist}^A(E, B))$.

In subsection 2.2, we show that two partial interpositions $K$ and $L$ whose boundaries have the same distance to A can be joined together to give a full A-interposition by means of a selecting set $S$.

## 2.1 Density of distances from A to the boundary of $C^{1,1}$ interpositions

For any closed non-empty $E, F ⊂ E$, then for any $σ ≥ 0$ such that $E$ and $F^{A,σ}$ are disjoint, it holds that

$$ \mathrm{dist}^A(E, F) = \mathrm{dist}^A(E, F^σ) + σ. \qquad (2.1) $$

Indeed, let $(p, q) ∈ E × F^σ$, there exists by definition a $q_σ ∈ F$ such that $|q - q_σ| ≤ σ$, as a consequence of the triangle inequality we find

$$ \begin{aligned} \|p - q\|^A + σ &\ge \|p - q_σ\|^A - \|q_σ - q\|^A + σ \\ &\ge \|p - q_σ\|^A \\ &\ge \mathrm{dist}^A(E, F). \end{aligned} $$

Taking the infimum over $(p, q) ∈ E × F^σ$, gives the inequality $\mathrm{dist}^A(E, F^σ) + σ ≥ \mathrm{dist}^A(E, F)$. For the other inequality, let $(p_0, q_0) ∈ E × F$ be a pair of points realizing $\mathrm{dist}^A(E, F)$, since $\mathrm{dist}^A(E, F) ≥ \mathrm{dist}^A(E, F^σ) > 0$, $p_0 ≠ q_0$ thus for all $ρ ∈ [0, σ]$,

$$ q_ρ := q_0 + ρ \frac{p_0 - q_0}{\|p_0 - q_0\|^A} $$

is well-defined and belongs to $F^σ$. Note that we must have $σ < \mathrm{dist}^A(E, F)$, otherwise $ρ_0 := \mathrm{dist}^A(E, F) ≤ σ$ and $q_{ρ_0} = p_0 ∈ E ∩ F^σ$ which contradicts the
assumption dist<sup>A</sup>(E, F<sup>σ</sup>) > 0. As a result the three points q<sub>0</sub>, q<sub>σ</sub> and p<sub>0</sub> are
aligned in this order. Hence because d<sup>A</sup> is euclidean we have,

$$
\begin{align*}
\mathrm{dist}^{\mathcal{A}}(E, F) &= \|q_{\sigma} - p_{0}\|^{\mathcal{A}} + \|q_{\sigma} - q_{0}\|^{\mathcal{A}} \\
&\geq \mathrm{dist}^{\mathcal{A}}(E, F^{\sigma}) + \sigma,
\end{align*}
$$

which ends our argument. This observation unlocks the following result.

**Lemma 2.1.** Let $A, B \subset \mathbb{E}$ be two closed non-empty and disjoint sets and assume that $\mathcal{K}$ is a $\mathbb{A}$-interposition between $A^{\mathcal{A},\sigma}$ and $B^{\mathcal{A},\rho}$ for some $\sigma, \rho \ge 0$, then $\mathcal{K}$ is a $\mathbb{A}$-interposition between $A$ and $B$.

Proof. It is obvious that (1.1) holds for $\mathcal{K}$ because $A \subset A^{\sigma}$ and $(B^{\rho})^{c} \subset B^{c}$.
Now we know from the assumption that

$$
\mathrm{dist}^{\mathcal{A}}(A^{\sigma}, B^{\rho}) = \mathrm{dist}^{\mathcal{A}}(A^{\sigma}, \partial \mathcal{K}) + \mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, B^{\rho}),
$$

and since all the terms above are positive by assumption, from (2.1), we find

$$
\mathrm{dist}^{\mathcal{A}}(A, B) - \sigma - \rho = \mathrm{dist}^{\mathcal{A}}(A, \partial \mathcal{K}) - \sigma + \mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, B) - \rho.
$$

in other words dist<sup>A</sup>(A, B) = dist<sup>A</sup>(A, ∂K) + dist<sup>A</sup>(∂K, B) and the result follows.

<figure><img src="image_4.png" /></figure>

As a direct corollary of this invariance of interpositions for tubular neigh-
borhoods, we find the following improvement of Theorem 1.1. The arguments
in the proof of this result will be invoked later for the proof of Theorem 1.5.

**Corollary 2.2.** Let $A, B \subset \mathbb{E}$ be two closed non-empty sets with $A$ compact, then there exists a dense set $\Sigma \subset (0, \mathrm{dist}^{\mathcal{A}}(A, B))$ such that for all $\sigma \in \Sigma$, there exists $K_{\sigma}$ with $C^{1,1}$ boundary which is a $\mathbb{A}$-interposition between $A$ and $B$ and such that

$$
\mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, A) = \sigma.
$$

Proof. Denote $\rho := \mathrm{dist}^{\mathcal{A}}(A, B)$, let $\sigma_0 \in (0, \rho)$, let $\eta > 0$ be small enough so that $\eta \le \min\{\sigma_0, \rho - \sigma_0\}$, and consider $A' := A^{\sigma_0 - \eta}$ and $B' := B^{(\rho - \sigma_0) - \eta}$. Since $A', B' \subset \mathbb{E}$ are closed disjoint and non-empty with $A'$ compact, thus from Theorem 1.1, there exists $\mathcal{K}$ a $C^{1,1}$ $\mathbb{A}$-interposition between $A'$ and $B'$. Now from Lemma 2.1 this implies that $\mathcal{K}$ is a $C^{1,1}$ $\mathbb{A}$-interposition between $A$ and $B$ and we have using (2.1),

$$
\begin{align*}
\mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, A) &= \mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, A^{\sigma_0 - \eta}) + \sigma_0 - \eta \\
&\geq \sigma_0 - \eta,
\end{align*}
$$

and

$$
\begin{align*}
\mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, A) &= \rho - \mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, B) \\
&= \rho - \mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, B^{\rho - \sigma_0 - \eta}) - \rho + \sigma_0 + \eta \\
&\leq \sigma_0 + \eta,
\end{align*}
$$

$$
\text{thus } \left| \mathrm{dist}^{\mathcal{A}}(\partial \mathcal{K}, A) - \sigma_0 \right| \le \eta \text{ and this ends our proof. } \square
$$

<figure><img src="image_5.png" /></figure>
## 2.2 Junction of partial interpositions

In the following we always consider that $A, B$ are closed subsets of the euclidean space $\mathbb{E}$ and $\mathcal{A}$ is some anisotropy operator on $\mathbb{E}$.

**Lemma 2.3.** Let $\mathcal{K}, \mathcal{L}, S \subset \mathbb{E}$ be three closed subsets and define

$$ \mathcal{J} := (\mathcal{K} \cap S) \cup (\mathcal{L} \cap S^c). $$

Assume that $A \subset \mathcal{J} \subset B^c$ and that there exists $\sigma \in (0, \text{dist}^{\mathcal{A}}(A, B))$ such that

$$ \begin{array}{rcl} & & \partial \mathcal{K} \cap \bar{S} \\ d^{\mathcal{A}}(x, A) & \ge & \sigma, \\ d^{\mathcal{A}}(x, B) & \ge & \text{dist}^{\mathcal{A}}(A, B) - \sigma, \quad \forall x \in \quad \begin{array}{c} \cup \\ \partial \mathcal{L} \cap \bar{S}^c \end{array} \\ & & \cup \\ & & \partial S \cap (K \cap \bar{L}^c \cup L \cap \bar{K}^c), \end{array} \quad (2.2) $$

then $\mathcal{J}$ is a $\mathcal{A}$-interposition between $A$ and $B$.

*Proof.* Since (2.2) translates (1.2), note that the proof is finished if we manage to show that

$$ \partial \mathcal{J} \subset (\partial \mathcal{K} \cap \bar{S}) \cup \partial \mathcal{L} \cap \bar{S}^c \cup (\partial S \cap (\mathcal{K} \cap \bar{\mathcal{L}}^c \cup \mathcal{L} \cap \bar{\mathcal{K}}^c)). $$

Using Moore's laws and because the closure distributes over finite unions, we have

$$ \begin{align*} \partial \mathcal{J} &= \overline{(\mathcal{K} \cap S) \cup (\mathcal{L} \cap S^c)} \cap \overline{((\mathcal{K} \cap S) \cup (\mathcal{L} \cap S^c))^c} \\ &= \overline{(\mathcal{K} \cap S) \cup \mathcal{L} \cap S^c} \cap \overline{(\mathcal{K} \cap S)^c \cap (\mathcal{L} \cap S^c)^c} \\ &= \overline{(\mathcal{K} \cap S) \cup \mathcal{L} \cap S^c} \cap \overline{(\mathcal{K}^c \cup S^c) \cap (\mathcal{L}^c \cup S^c)}, \\ &=: P(\mathcal{K}, \mathcal{L}, S) \cup P(\mathcal{L}, \mathcal{K}, S^c) \end{align*} $$

where we denoted $P(\mathcal{K}, \mathcal{L}, S) := \overline{\mathcal{K} \cap S} \cap \overline{(\mathcal{K}^c \cup S^c) \cap (\mathcal{L}^c \cup S)}$. Observe now that,

$$ \begin{align} \overline{(\mathcal{K}^c \cup S^c) \cap (\mathcal{L}^c \cup S)} &= \overline{(\mathcal{K}^c \cap \mathcal{L}^c) \cup (\mathcal{K}^c \cap S) \cup (\mathcal{L}^c \cap S^c)} \nonumber \\ &= \overline{\mathcal{K}^c \cap \mathcal{L}^c} \cup \overline{\mathcal{K}^c \cap S} \cup \overline{\mathcal{L}^c \cap S^c}. \tag{2.3} \end{align} $$

Now using the fact that $\overline{E \cap F} \subset \overline{E} \cap \overline{F}$, after using (2.3), we obtain

$$ \begin{align*} P(\mathcal{K}, \mathcal{L}, S) &\subset (\partial \mathcal{K} \cap \bar{S} \cap \bar{\mathcal{L}}^c) \cup (\partial \mathcal{K} \cap \bar{S}) \cup (\partial S \cap \mathcal{K} \cap \bar{\mathcal{L}}^c) \\ &\subset (\partial \mathcal{K} \cap \bar{S}) \cup (\partial S \cap \mathcal{K} \cap \bar{\mathcal{L}}^c), \end{align*} $$

and the result falls at once. $\square$

The lemma is useful to build new interpositions from two interpositions $\mathcal{K}$ and $\mathcal{L}$ whose boundaries have a common distance with $A$. It can also be used to locally replace part of an interposition $\mathcal{L}$ with a suitable new surface.
**Corollary 2.4.** Let $K, \mathcal{L}$ be two $d^\mathcal{A}$-interpositions between A and B such that

$$d^{\mathcal{A}}(A, \partial K) = d^{\mathcal{A}}(A, \partial \mathcal{L}),$$

then for any $S \subset \mathbb{E}$, the junction $\mathcal{J} := (K \cap S) \cup (\mathcal{L} \cap S^c)$ is a $\mathbb{A}$-interposition between A and B.

**Corollary 2.5.** Let $K, \mathcal{L}, S$ be closed subsets of $\mathbb{E}$ such that $K$ is a $\mathbb{A}$-interposition between A and B and assume that

$$A \cap S \subset \mathcal{L} \subset B^c \cap S$$

and that there exists $\sigma \in (0, \text{dist}^{\mathcal{A}}(A, B))$ such that

$$\begin{array}{l} d^{\mathcal{A}}(x, A) \geq \sigma \\ d^{\mathcal{A}}(x, B^c) \geq d^{\mathcal{A}}(A, B) - \sigma \quad \forall x \in \partial \mathcal{L} \cap S, \end{array}$$

then $\mathcal{J} = K \cap S^c \cup \mathcal{L} \cap S$ is a $\mathbb{A}$-interposition between A and B.

# 3 Proof of Theorem 1.5

The proof follows closely the lines of [6] and [2, Theorem 12.1], making the needed modifications along the way.

We provide here an intermediate result, whose proof can be adapted from [6, Lemma 5.1] or [2, Theorem 12.4] by means of a linear transformation. We will see it provides a $C^{1,1}$ level-set representation of interpositions but only near the critical points of A and B.

**Lemma 3.1.** Let $\varepsilon > 0$, $A, B \subset \mathbb{E}$ be closed and disjoint with A compact and

$$\phi(p) := d^{\varepsilon}(p, (A^{\varepsilon, \alpha})^c),$$

for some $\alpha \in (0, \text{dist}^{\varepsilon}(A, B))$, then for all $p_0 \in \mathcal{D}^{\varepsilon}(A, B) \cap A^{\varepsilon, \alpha} \cap A^c$, there exists a neighborhood $V$ of $p_0$ such that $\phi \in C^{1,1}(V)$.

Let $0 < \varepsilon < \varepsilon_\Gamma(A, B)$, where $\varepsilon_\Gamma$ is defined in (1.5) of $\varepsilon_\Gamma$. As a first remark, note that we only to construct one non-characteristic $\varepsilon$-interposition $K_\varepsilon$ between A and B. Indeed because A and B are arbitrary, the density of $\Sigma_\varepsilon \subset (0, \text{dist}^\varepsilon(A, B))$ will follow by applying the same line of arguments as in the proof of Corollary 2.2 and by noting that, since $\mathcal{D}^\varepsilon(A^{\varepsilon, \sigma}, B^{\varepsilon, \rho}) = \mathcal{D}^\varepsilon(A, B) \cap (A^{\varepsilon, \sigma})^c \cap (B^{\varepsilon, \rho})^c$ for any $\sigma, \rho \ge 0$, a non-characteristic interposition between $A^{\varepsilon, \sigma}$ and $B^{\varepsilon, \rho}$ is also be a non-characteristic interposition between A and B.

From the assumption we have $0 < \text{dist}^0(A, B) < +\infty$, thus it holds that $\text{dist}^\varepsilon(A, B) > 0$. Assume, without loss of generality, that $\text{dist}^\varepsilon(A, B) = 1$. We fix an arbitrary $\alpha \in (0, 1)$ and consider $A^{\varepsilon, \alpha}$ the $\alpha$-neighborhood of A and the function

$$\phi(p) := d^{\varepsilon}(p, (A^{\varepsilon, \alpha})^c).$$
The set $\mathcal{R}_\alpha := A^{\epsilon, \alpha} \cap A^c$ is the open region lying between the boundaries of $A^{\epsilon, \alpha}$ and $A$. It is clear that for any $p \in \mathcal{R}_\alpha$, the following inequalities

$$1 - d^{\epsilon}(p, B) \leq \alpha - \phi(p) \leq d^{\epsilon}(p, A). \qquad (3.1)$$

hold, as a consequence it is clear that for any $\sigma \in (0, \alpha)$, the $\sigma$-sublevel set of $\phi$

$$C_{\sigma} := \{p \in \mathbb{R} \times \mathbb{R}^N \mid \phi(p) \geq \sigma\},$$

is a $\epsilon$-interposition between $A$ and $B$ because $A \subset\subset C_\sigma \subset\subset B^c$.

Observe however that $\partial C_\sigma$ does not have the wanted $C^{1,1}$ regularity, thus a careful regularization of $\phi$ is constructed in the next steps.

* Lemma 3.1 implies that for any $p_0 \in \mathcal{D}^{\epsilon}(A, B) \cap \mathcal{R}_{\alpha}$, there exists an open neighborhood $V \subset \mathcal{R}_{\alpha}$ of $p_0$ such that $\phi$ is $C^{1,1}$ on $V$.

**Claim 1:** For any $p_0 \in \mathcal{D}^{\epsilon}(A, B) \cap \mathcal{R}_{\alpha}$ we also have

$$\nabla_x \phi(p_0) \neq 0.$$

Indeed remark that $d^{\epsilon}(\cdot, A/B) = d_1(R_{\epsilon}, A_{\epsilon}/B_{\epsilon})$, where $R_{\epsilon}: p = (t, x) \mapsto (t/\epsilon, x)$ and $A_{\epsilon}/B_{\epsilon} = R_{\epsilon}(A)/R_{\epsilon}(B)$. As a result $R_{\epsilon}(\mathcal{D}^{\epsilon}(A, B)) = \mathcal{D}(A_{\epsilon}, B_{\epsilon})$, hence $p_{0,\epsilon} := R_{\epsilon}p_0 \in \mathcal{D}(A_{\epsilon}, B_{\epsilon})$.

Denote the set of $d^{\epsilon}$-projections of $p_0$ on $A$ and $B$ respectively as $\Pi_{\epsilon}(p_0, A)$ and $\Pi_{\epsilon}(p_0, B)$ and note that since $p_0 \in \mathcal{D}^{\epsilon}(A, B)$ and from the equality case of the triangle inequality for the euclidean metric $d^{\epsilon}$, that for all $p_A \in \Pi_{\epsilon}(p_0, A)$ and $p_B \in \Pi_{\epsilon}(p_0, B)$, we have

$$p_0 = \lambda p_A + (1 - \lambda) p_B,$$

for some $\lambda \in (0, 1)$. Consequently it must holds that $\Pi_{\epsilon}(p_0, A) = \{p_A = (t_A, x_A)\}$ and $\Pi_{\epsilon}(p_0, B) = \{p_B = (t_B, x_B)\}$. This implies that $\Pi(p_{0,\epsilon}, A_{\epsilon}) = \{R_{\epsilon}p_A\}$ and $\Pi(p_{0,\epsilon}, B_{\epsilon}) = \{R_{\epsilon}p_B\}$; $\Pi(p_{0,\epsilon}, A_{\epsilon})$ being single-valued, $d(\cdot, A_{\epsilon})$ is differentiable at $p_{0,\epsilon}$ (see e.g. [13, Theorem 3.3 (iii) & (vi)]) and its differential is

$$Dd(\cdot, A_{\epsilon})|_{p=p_{0,\epsilon}} = \begin{pmatrix} \frac{t_B - t_A}{\epsilon} \\ x_B - x_A \end{pmatrix}.$$

Now since $d^{\epsilon}(p, A) = d(R_{\epsilon}p, A_{\epsilon})$, $d^{\epsilon}$ is also differentiable at $p_0$ and from the chain rule we obtain

$$Dd^{\epsilon}(\cdot, A)|_{p=p_0} = \begin{pmatrix} \frac{t_B - t_A}{\epsilon^2} \\ x_B - x_A \end{pmatrix}.$$

Also we infer from (3.1) that $\alpha - \phi$ touches $d^{\epsilon}(\cdot, A)$ from below at $p_0$, therefore $D\phi(p_0)$ coincides with $Dd^{\epsilon}(\cdot, A)|_{p=p_0}$. Now from the triangle inequality it is clear that $(p_A, p_B) \in \Gamma_{\epsilon}(A, B)$, and since $0 < \epsilon < \epsilon_{\Gamma}$ we have $x_A \neq x_B$ and we can conclude that $\nabla_x \phi(p_0) \neq 0$.
• Denote

$$ U := \left\{ p \in \mathcal{R}_{\alpha} \mid \frac{\alpha}{3} < d^{\varepsilon}(p, A) < \frac{2\alpha}{3} \right\}, $$

a bounded open set with $U \subset \mathcal{R}_{\alpha}$.

**Claim 2:** There exists $\eta > 0$ and an open set $O \subset \mathcal{R}_{\alpha}$ such that

$$ \mathcal{D}^{\varepsilon}(A, B) \cap \bar{U} \subset \mathcal{O}, $$

the function $\phi$ is $C^{1,1}$ in $O$ and

$$ |\nabla_x \phi| \geq \eta \quad \text{on } O. \qquad (3.2) $$

Note indeed that $\mathcal{D}^{\varepsilon}(A, B) \cap \bar{U}$ is a compact set, $\nabla_x \phi$ is continuous and does not vanish on $\mathcal{D}^{\varepsilon}(A, B)$ from **Claim 1**. Hence there exists an open set $O \subset \mathcal{R}_{\alpha}$ containing $\mathcal{D}^{\varepsilon}(A, B) \cap \bar{U}$ on which

$$ |\nabla_x \phi| \geq \eta. $$

• Fix an open set $O'$ satisfying

$$ \mathcal{D}^{\varepsilon}(A, B) \cap \bar{U} \subset O' \subset \bar{O}' \subset O. \qquad (3.3) $$

The following lemma gives us a carefully crafted regularization of $\phi$, the proof of which is postponed to subsection 3.1.

**Lemma 3.2.** There exists a function $\psi : \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ such that

(i) $\psi \in C^{1,1}(U)$ and $\psi \in C^{\infty}(\mathbb{R} \times \mathbb{R}^N \setminus O)$,

(ii) $\psi(p) = \phi(p)$ for all $p \in O'$,

(iii) $\|\psi - \phi\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} \le \frac{\alpha}{12}$,

(iv) $\|\nabla_x \psi - \nabla_x \phi\|_{L^{\infty}(O)} \le \frac{\eta}{2}$,

(v) $1 - d^{\varepsilon}(p, B) \le \alpha - \psi(p) \le d^{\varepsilon}(q, A)$ for all $q \in \bar{U}$,

(vi) $1 - d^{\varepsilon}(q, B) < \alpha - \psi(p) < d^{\varepsilon}(p, A)$ for all $q \in \bar{U} \setminus \bar{O}'$.

Now that $\psi$ is constructed, we construct in the following steps the interpolation $K_{\varepsilon}$.

• Denote $I_{\alpha} = [\frac{5}{12}\alpha, \frac{7}{12}\alpha]$ and note that by virtue of Lemma 3.3 - (iii),

$$ \forall \sigma \in I_{\alpha}, \quad \Lambda(\psi, \sigma) := \{p \in \mathbb{R} \times \mathbb{R}^N \mid \psi(p) = \sigma\} \subset U. \qquad (3.4) $$

By Sard's lemma, since $\psi \in C^{\infty}(\mathbb{R} \times \mathbb{R}^N \setminus O)$ and since, by virtue of Lemma 3.3 - (iv) and (3.2), $\nabla_x \psi \ne 0$ in $\bar{O}$, there exists a full-measure subset $C \subset I_{\alpha}$ such that for any $\sigma \in C$, $\sigma$ is not a critical value of $\psi$, i.e.
$D\psi(p) \neq 0$ for any $p \in \Lambda(\psi, \sigma)$. Note that therefore $C \neq \emptyset$ and is dense in $I_\alpha$.

For any $t \in \mathbb{R}$ we also consider the map,

$$ \psi(t, \cdot) : \begin{cases} \mathbb{R}^N & \to \mathbb{R} \\ x & \mapsto \psi(t, x). \end{cases} $$

Again from Sard's lemma, since $\psi(t, \cdot) \in C^\infty(\mathbb{R}^N \setminus O(t))$ and $\nabla_x \psi \neq 0$ on $O(t)$, there exists a full-measure subset $C_t \subset I_\alpha$ of noncritical values of $x \mapsto \psi(t, x)$.

Fix some $\sigma_0 \in C \cap [\frac{11}{24}\alpha, \frac{13}{24}\alpha]$. Note that due to (3.2) and Lemma 3.3 - (iv), $\sigma_0$ is a noncritical value of $\psi(t, \cdot)$ inside $O(t)$, but it might be critical outside $O(t)$, hence we will introduce a spatial perturbation of $\psi(t, \cdot)$.

• Denote $V, V'$ two open sets such that

$$ O' \subset V' \subset V \subset O. $$

From Lemma 3.3 - (vi), there exists $\lambda > 0$ such that

$$ \min_{p \in \bar{U} \setminus V'} f_A(p) - (\alpha - \psi(p)) \geq \lambda \quad (3.5) $$

and

$$ \min_{p \in \bar{U} \setminus V'} (\alpha - \psi(p)) - f_B(q) \geq \lambda. \quad (3.6) $$

Take some cut-off function $\xi \in C^\infty(\mathbb{R} \times \mathbb{R}^N)$ such that $0 \le \xi \le 1$ and

$$ \xi(p) = \begin{cases} 0 & \text{for } p \in \bar{V} \\ 1 & \text{for } p \in O^c \end{cases} $$

Let $t \in \mathbb{R}$ be fixed. By density of $C_t$ in $I_\alpha$ we can choose a $\sigma_t \in C_t$ such that

$$ |\sigma_0 - \sigma_t| \le \min \left\{ \frac{\alpha}{96}, \frac{\eta}{4 \|\nabla_x \xi\|_{\infty}}, \frac{\lambda}{2} \right\}. \quad (3.7) $$

Define for $x \in \mathbb{R}^N$ the purely spatial perturbation

$$ k^t(x) := (\sigma_0 - \sigma_t)\xi(t, x) \quad (3.8) $$

and the perturbed function $\psi^t(s, x) \mapsto \psi(s, x) + k^t(x)$.

• Note first from (3.4) that for all $\sigma \in \mathbb{R}$ such that $|\sigma - \sigma_0| \le \alpha/48$, it holds that

$$ \Lambda(\psi^t, \sigma) \subset \bigcup_{|\sigma' - \sigma_0| \le \frac{\alpha}{24}} \Lambda(\psi, \sigma') \subset U. \quad (3.9) $$
Observe also that $\sigma_0$ is a noncritical value of $\psi^t(t, \cdot)$. Indeed by the definition (3.8) of $k^t$, we have

$$ \Lambda(\psi^t(t, \cdot), \sigma_0) \setminus \mathcal{O}(t) = \Lambda(\psi(t, \cdot), \sigma_t) \setminus \mathcal{O}(t) $$

and $\sigma_0$ is a noncritical value of $\psi(t, \cdot)$. Now if $x \in \Lambda(\psi^t(t, \cdot), \sigma_0) \cap \mathcal{O}(t)$, then using (3.2), Lemma 3.3 - (iv) and (3.7) we find,

$$ \begin{aligned} |\nabla_x \psi^t(t, x)| &\ge |\nabla_x \psi(t, x)| - |\nabla_x k^t(x)| \\ &\ge |\nabla_x \phi(t, x)| - |\nabla_x \psi(t, x) - \nabla_x \phi(t, x)| \\ &\quad - |\sigma_0 - \sigma_t||\nabla_x \xi(t, x)| \\ &\ge \eta - \frac{\eta}{2} - \frac{\eta}{4} \\ &= \frac{\eta}{4} > 0. \end{aligned} $$

* **Claim 3:** There exists $\delta_t^W > 0$ and an open set $W_t \subset U$ such that for all $\sigma \in \mathbb{R}$ with $|\sigma - \sigma_0| \le \frac{\alpha}{48}$, we have

$$ \Lambda(\psi^t, \sigma) \cap ([t - \delta_t^W, t + \delta_t^W] \times \mathbb{R}^N) \subset W_t \quad (3.10) $$

and for all $p \in W$,

$$ f_A(p) \le \alpha - \psi^t(p) \le f_B(p). \quad (3.11) $$

Indeed, since $V' \subset V$, there exists an open set $Y \subset \mathbb{R}^N$ and $\delta_t^W > 0$ such that for all $s \in (t - \delta_t^W, t + \delta_t^W)$,

$$ V'(s) \subset Y \subset V(s). \quad (3.12) $$

Denoting $Y_\delta := (t - \delta_t^W, t + \delta_t^W) \times Y$, we can split for any $\sigma \in \mathbb{R}$ with $|\sigma - \sigma_0| \le \frac{\alpha}{48}$, $\Lambda(\psi^t, \sigma)$ into two parts as follows

$$ \begin{aligned} \Lambda(\psi^t, \sigma) &= (\Lambda(\psi^t, \sigma) \cap Y_\delta) \cup (\Lambda(\psi^t, \sigma) \setminus Y_\delta) \\ &=: P_1(\sigma) \cup P_2(\sigma). \end{aligned} $$

From (3.9) we know $P_1(\sigma) \subset Y_\delta \cap U$ and since $k^t$ is supported outside of $\bar{V}$ we have $\psi_t = \psi$ on $Y_\delta \cap U$. Since $\psi$ satisfies (v) of Lemma 3.3, we conclude that (3.11) holds for all $p \in Y_\delta \cap U$.

Also from (3.9) we know that $P_2(\sigma) \subset U$ and remark that by (3.12) and (3.5)-(3.6), since $Y_\delta \subset V$, we have for all $p \in P_2(\sigma)$,

$$ f_B(p) + \lambda - |\sigma_0 - \sigma_t| \le \alpha - \psi^t(p) \le f_A(p) - \lambda + |\sigma_0 - \sigma_t|. $$

Thus by our choice of $\sigma_t$ (3.7),

$$ f_B(p) + \frac{\lambda}{2} \le \alpha - \psi^t(p) \le f_A(p) - \frac{\lambda}{2} $$
Since $\psi^t$ is $C^1$ and vanishes outside of a bounded set it is $L$-Lipschitz continuous for some $L > 0$. Thus for each $p \in P_2(\sigma) + \mathbb{B}_{N+1}(0, \frac{\lambda}{4L})$, the inequalities (3.11) hold. It then suffices to take $W_t$ as the open set

$$ \left( Y_{\delta} \cup \bigcup_{|\sigma - \sigma_0| \le \frac{\lambda}{4\delta}} \left( P_2(\sigma) + \mathbb{B}_{N+1}\left(0, \frac{\lambda}{4L}\right) \right) \right) \cap U $$

to obtain **Claim 3**.

* By continuity of $(s, x) \mapsto \nabla_x \psi^t(s, x)$, since $\sigma_0$ is a noncritical value of $x \mapsto \psi^t(t, x)$, there exists $\delta_t^Z > 0$ and an open set $Z_t \subset \mathbb{R} \times \mathbb{R}^N$ such that

$$ \Lambda(\psi^t, \sigma_0) \cap \left( (t - \delta_t^Z, t + \delta_t^Z) \times \mathbb{R}^N \right) \subset Z_t \quad (3.13) $$

and such that $\nabla_x \psi^t \neq 0$ on $Z_t$. Consequently from (3.10) and (3.13)

$$ \Lambda(\psi^t, \sigma_0) \cap \left( (t - \delta_t, t + \delta_t) \times \mathbb{R}^N \right) \subset W_t \cap Z_t, $$

where $\delta_t := \min \delta_t^W, \delta_t^Z$.

* We are now ready to construct the interposition. Since $t \in \mathbb{R}$ was arbitrary, we can find points $\{\tau_i\}_{i \in \mathbb{Z}}$, $\{t_i\}_{i \in \mathbb{Z}} \subset \mathbb{R}$ with $t_i < \tau_i < t_{i+1}$, such that

$$ \mathbb{R} = \bigcup_{i \in \mathbb{Z}} [t_i, t_{i+1}] =: \bigcup_{i \in \mathbb{Z}} J_i $$

and

$$ |\tau_i - t_{i+1}|, |\tau_i - t_i| < \delta_{\tau_i}, \quad \forall 1 \le i \le N \quad (3.14) $$

as well as a corresponding family of functions $\psi^i := \psi^{\tau_i} \in C^{1,1}(J_i \times \mathbb{R}^N)$ such that,

$$ \Lambda(\psi^i, \sigma_0) \subset W_{\tau_i} \cap Z_{\tau_i}. \quad (3.15) $$

Define then for each $i \in \mathbb{Z}$, the closed set

$$ \mathcal{K}_i := \{ p \in J_i \times \mathbb{R}^N \mid \psi^i(p) \ge \sigma_0 \}. $$

Remark that since $\sigma_0$ is a spatially non-critical value for $\psi^i$ on $J_i \times \mathbb{R}^N$, it holds that $\mathcal{K}_i \in C_{t,pw}^{1,1}(J_i)$ and we have the representation

$$ \partial \mathcal{K}_i = \partial_{J_i} \mathcal{K}_i \cup \left( \{t_i\} \times \{\psi^i(t_i, \cdot) > \sigma_0\} \right) \cup \left( \{t_{i+1}\} \times \{\psi^i(t_{i+1}, \cdot) > \sigma_0\} \right). \quad (3.16) $$

We now define our candidate interposition as

$$ \mathcal{K}_{\varepsilon} := \bigcup_{i \in \mathbb{Z}} \mathcal{K}_i \quad \in C_{t,pw}^{1,1}(\mathbb{R}). $$
• We here show that $\mathcal{K}_\varepsilon$ is a $\varepsilon$-interposition between $A$ and $B$. From Lemma 3.3 - (iii) and the definition of each $\psi^i$, we have $C_{\frac{4}{12}\alpha} \subset \mathcal{K}_\varepsilon \subset C_{\frac{s}{12}\alpha}$ therefore $A \subset\subset \mathcal{K}_\varepsilon \subset\subset B^c$.

Note that $\mathcal{K}_\varepsilon$ is closed since $\mathcal{K}_\varepsilon \cap B$ is closed for all closed and bounded $B \subset \mathbb{R} \times \mathbb{R}^N$ and thus we can write

$$ \begin{aligned} \partial \mathcal{K}_\varepsilon &= \bigcup_{i \in \mathbb{Z}} \left( \mathcal{K}_i \cap \overline{\bigcap_{j \in \mathbb{Z}} \mathcal{K}_j^c} \right) \\ &= \bigcup_{i \in \mathbb{Z}} \left( \mathcal{K}_i \cap \overline{\bigcap_{|j-i| \le 1} \mathcal{K}_j^c} \right) \\ &= \bigcup_{i \in \mathbb{Z}} \left( \partial \mathcal{K}_i \cap \overline{\bigcap_{|j-i| \le 1} \mathcal{K}_j^c} \right) \end{aligned} \quad (3.17) $$

where we used that $\mathcal{K}_i \subset \bigcap_{|j-i| \ge 2} \mathcal{K}_j^c$ and that $\mathcal{K}_i^c \cap \overline{\mathcal{K}_i^c} = \emptyset$.

Let now $p = (t, x) \in \partial \mathcal{K}_\varepsilon$, there exists some $i \in \mathbb{Z}$ such that

$$ p \in \partial \mathcal{K}_i \cap \overline{\bigcap_{|j-i| \le 1} \mathcal{K}_j^c}. $$

Using (3.16), either $p \in \partial J_i \mathcal{K}_i$ and thus $\psi^i(p) = \sigma_0$ hence from (3.15) $p \in W_{\tau_i}$ and the interposition relation holds at $p$ by virtue of (3.11). Otherwise, say $t = t_{i+1}$, then $\psi^i(p) > \sigma_0$ and $p \in \overline{\mathcal{K}_i^c \cap \mathcal{K}_{i+1}^c}$, this last condition implies

$$ \psi^i(p) \le \sigma_0 \quad \text{or} \quad \psi^{i+1}(p) \le \sigma_0. $$

However since $\psi^i(p) > \sigma_0$, we must conclude that $\psi^{i+1}(p) \le \sigma_0$. Now using $|\sigma_0 - \sigma_{\tau_i}|, |\sigma_0 - \sigma_{\tau_{i+1}}| \le \frac{\alpha}{96}$, we deduce that

$$ \begin{aligned} |\psi^i(p) - \sigma_0| &= \psi^i(p) - \sigma_0 \\ &= \psi(p) + k^{\tau_i}(x) - \sigma_0 \\ &= \psi^{i+1}(p) - \sigma_0 + k^{\tau_i}(x) - k^{\tau_{i+1}}(x) \\ &\le |\sigma_{\tau_i} - \sigma_{\tau_{i+1}}| \\ &\le \frac{\alpha}{48}. \end{aligned} $$

Hence from (3.14) and (3.10), we conclude that $p \in W_{\tau_i}$, thus (3.11) also holds and allows to conclude.

• We finally show that $\mathcal{K}_\varepsilon$ is a non-characteristic $\varepsilon$-interposition. Let $p \in \partial \mathcal{K}_\varepsilon \cap \mathcal{D}^\varepsilon(A, B)$. We only need to check the non-characteristic conditions
in the case $p = (t_i, x)$ for some $i \in \mathbb{Z}$, because we already know that $\partial_{\phi} K_i$
is locally $C^{1,1}$ around $p$ with finite velocity.

Since $\partial K_\varepsilon \subset U$, we know from (3.3) that $p \in \mathcal{D}^\varepsilon(A, B) \cap U \subset \mathcal{O}'$.
Therefore from Lemma 3.3 (iii) and the definitions of $\psi^i, \psi^{i+1}$ it holds
that $\psi^i = \psi^{i+1} = \psi = \phi$ in some neighborhood of $p$, also from (3.17)
$\phi - \sigma_0 \in C^{1,1}(\mathcal{O}')$ is a spatially non-degenerate level-set representation of
$\partial K_\varepsilon$ around $p$.

□

Remark 3.1. The following remark is useful for applications. Given two points $(p, q) \in \Gamma_{\varepsilon}(A, B)$, with $\varepsilon \in (0, \varepsilon_{\Gamma}(A, B))$, the proof above gives us the possibility to choose an interposition $K_{\varepsilon} \in C_{t,pw}^{1,1}(\mathbb{R})$, which will then depend on this pair $(p, q)$ such that the critical point $z = (t, x) \in [p, q] \cap \partial K$ does not lie on a time of discontinuity for $K$, i.e. $t \notin \{t_i\}_i$.

**3.1 Proof of Lemma 3.3**

**Lemma 3.3.** There exists a function $\psi : \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ such that

(i) $\psi \in C^{1,1}(U)$ and $\psi \in C^{\infty}(\mathbb{R} \times \mathbb{R}^N \setminus \mathcal{O})$,

(ii) $\psi(q) = \phi(q)$ for all $q \in \mathcal{O}'$,

(iii) $\|\psi - \phi\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} \le \frac{\alpha}{12},$

(iv) $\|\nabla_x \psi - \nabla_x \phi\|_{L^{\infty}(\mathcal{O})} \le \frac{\eta}{2},$

(v) $1 - d^{\varepsilon}(q, B) \le \alpha - \psi(q) \le d^{\varepsilon}(q, A)$ for all $q \in \bar{U}$,

(vi) $1 - d^{\varepsilon}(q, B) < \alpha - \psi(q) < d^{\varepsilon}(q, A)$ for all $q \in \bar{U} \setminus \bar{\mathcal{O}}'$.

*Proof.* Fix a cut-off function $\theta \in C_c^\infty(\mathbb{R} \times \mathbb{R}^N)$, $0 \le \theta \le 1$ such that

$$
\theta(q) = \begin{cases} 1 & \text{if } q \in \overline{\mathcal{O}'} \\ 0 & \text{if } q \notin \mathcal{O}' \end{cases} \quad \text{and } \theta(q) < 1 \text{ for } q \notin \overline{\mathcal{O}'} \tag{3.18}
$$

as well as a mollifier $\rho \in C_c^\infty(\mathbb{R} \times \mathbb{R}^N)$ such that,

$$
\begin{equation} \tag{3.19}
\begin{split}
\operatorname{supp} \rho \subset \mathbb{B}_1 &:= \mathbb{B}_{N+1}(0, 1), \quad \rho \ge 0, \\
\int_{\mathbb{R} \times \mathbb{R}^N} \rho(p) \, dp &= 1.
\end{split}
\end{equation}
$$

* Let $\delta > 0$ and $h \in C^\infty(\mathbb{R} \times \mathbb{R}^N)$. Define $\phi_\delta := \phi * \rho_\delta \in C^\infty(\mathbb{R} \times \mathbb{R}^N)$ the mollification of $\phi$ with $\rho_\delta(p) := \frac{1}{\delta}\rho(\frac{p}{\delta})$ and define the candidate

$$
\psi_{\delta,h} := \theta\phi + (1 - \theta)(\phi_{\delta} - h). \quad (3.20)
$$
• Note that since $\phi \in C^{1,1}(O)$, $\psi_{\delta,h}$ already satisfies (i)-(ii). Moreover since $\phi$ is globally Lipschitz on $\mathbb{R} \times \mathbb{R}^N$ and $\nabla_x \phi$ is globally Lipschitz on $O$, the following uniform convergences hold,

$$ \begin{aligned} \|\phi_{\delta} - \phi\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} & \xrightarrow{\delta \to 0^+} 0, \\ \|\nabla_x \phi_{\delta} - \nabla_x \phi\|_{L^{\infty}(O)} & \xrightarrow{\delta \to 0^+} 0. \end{aligned} \quad (3.21) $$

Also we have the following straightforward estimate,

$$ \|\phi - \psi_{\delta,h}\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} \le \|\phi - \phi_{\delta}\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} + \|h\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)}. \quad (3.22) $$

Now in $O$, $\phi \in C^{1,1}$ hence from (3.20) we have

$$ \nabla_x \psi_{\delta,h} - \nabla_x \phi = \nabla_x \theta(\phi - \phi_{\delta}) + (1 - \theta)(\nabla_x \phi_{\delta} - \nabla_x \phi - \nabla_x h), $$

and thus

$$ \begin{aligned} \|\nabla_x \psi_{\delta,h} - \nabla_x \phi\|_{L^{\infty}(O)} \le \|\nabla_x \theta\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} \|\phi - \phi_{\delta}\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} \\ + \|\nabla_x \phi - \nabla_x \phi_{\delta}\|_{L^{\infty}(O)} + \|\nabla_x h\|_{L^{\infty}(\mathbb{R} \times \mathbb{R}^N)} \end{aligned} \quad (3.23) $$

Remark that $\psi_{\delta,h}$ readily satisfies inequalities (v) in $O'$ since $\psi_{\delta,h} = \phi$ in $O'$.

For notational convenience, denote for $\delta > 0$ and $p \in \mathbb{R} \times \mathbb{R}^N$,

$$ \begin{aligned} f_A(p) &:= d^\epsilon(p, A), & g_{A,\delta} &:= f_A - (\alpha - \phi_\delta), \\ f_B(p) &:= 1 - d^\epsilon(p, B) & g_{B,\delta} &:= f_B - (\alpha - \phi_\delta) \end{aligned} $$

which are $\frac{2}{\epsilon}$-Lipschitz continuous functions.

• Our goal is to construct, for each $\delta > 0$ small enough, a function $h_\delta \in C^\infty(\mathbb{R} \times \mathbb{R}^N)$ which satisfies the following strict inequalities

$$ g_{B,\delta}(p) < h_\delta(p) < g_{A,\delta}(p), \quad \text{for all } p \in \bar{U} \setminus O', \quad (3.24) $$

as well as the following bounds,

$$ \begin{aligned} \|h_\delta\|_{L^\infty(\mathbb{R} \times \mathbb{R}^N)} &\le \frac{\alpha}{24}, \\ \|\nabla_x h_\delta\|_{L^\infty(U)} &\le \frac{\eta}{4}. \end{aligned} \quad (3.25) $$

If we can construct such a family of functions, we will be done. Indeed from the definition (3.18) of $\theta$, (3.24) will imply (v)-(vi) for $\psi_{\delta,h_\delta}$, and for $\delta > 0$ small enough, (3.25) combined with (3.21) and (3.22)-(3.23) will imply (iii)-(iv) for $\psi_{\delta,h_\delta}$.
• By continuity of $f_A$, $f_B$ and since $f_A > f_B$ on the compact set $K = \overline{U} \setminus O'$, there exists $\kappa > 0$, such that

$$ \min_{q \in K} \{f_A(q) - f_B(q)\} \geq \kappa > 0. \qquad (3.26) $$

Note also that $g_{A,\delta} - g_{B,\delta} = f_A - f_B \geq \kappa$ for any $\delta > 0$. Now by virtue of (3.1), it holds that $f_A \geq \alpha - \phi$ and $f_B \leq \alpha - \phi$ in $\mathcal{R}_\alpha$. Denote $U_* := U + \mathbb{B}_{R_*} R_*) \subset \mathcal{R}_\alpha$ for some $R_* > 0$ fixed and small enough, and define for all $\delta > 0$ the two nonnegative quantities

$$ M_A(\delta) := \left( \inf_{p \in U_*} g_{A,\delta}(p) \right)_- \quad \text{and} \quad M_B(\delta) := \left( \sup_{p \in U_*} g_{B,\delta}(p) \right)_+ . \qquad (3.27) $$

The quantity $M_A(\delta)$ (resp. $M_B(\delta)$) measures by how much the inequality $f_A \geq \alpha - \phi_\delta$ (resp. $f_B \leq \alpha - \phi_\delta$) fails. Remark that since $(\phi_\delta)_{\delta>0}$ converges uniformly to $\phi$ in $\mathcal{R}_\alpha$, we have $M_A(\delta), M_B(\delta) \to 0$ as $\delta \to 0$.

Let us define for $\delta > 0$ the following related quantities

$$ r(\delta) := (\max\{M_A(\delta), M_B(\delta)\})^{\frac{1}{2}}, \qquad (3.28) $$

$$ R(\delta) := (\max\{M_A(\delta), M_B(\delta)\})^{\frac{1}{2}}, \qquad (3.29) $$

and observe that $r(\delta), R(\delta) \to 0$ as $\delta \to 0$.

• Let $\delta > 0$, we will now construct $h_\delta$ as the sum of two parts: $h_{A,\delta}$ and $h_{B,\delta}$, which will each satisfy the inequalities (3.24) on suitable subsets of $K$. We here construct $h_{A,\delta}$ and then explain how to obtain $h_{B,\delta}$.

Denote

$$ U_{A,\delta} := \{q \in K \mid g_{A,\delta} > 0\}, \qquad (3.30) $$

which is a relative open subset of $K$ and

$$ K_{B,\delta} := \{q \in K \mid g_{B,\delta} \geq 0\} $$

which is a compact subset of $K$. By virtue of (3.26), $U_{A,\delta}$ contains $K_{B,\delta}$ and since the family of functions $\mathcal{F} := \{g_{A,\delta}\}_{\delta>0} \cup \{g_{B,\delta}\}_{\delta>0}$ is equi-Lipschitz (with constant $L_\varepsilon := 2/\varepsilon$), this implies further that there exists $R_{**} := \frac{\kappa}{2L_\varepsilon} > 0$, such that for all $\delta > 0$,

$$ K \cap (K_{B,\delta} + \mathbb{B}_{R_{**}} R_{**})) \subset U_{A,\delta}. \qquad (3.31) $$

Now there exists $\zeta_\delta \in C_c^\infty(\mathbb{R} \times \mathbb{R}^N)$ such that $0 \leq \zeta_\delta \leq 1$, $\zeta_\delta = 1$ on $K_{B,\delta}$, $\zeta_\delta = 0$ outside of $K_{B,\delta} + \mathbb{B}_{R(\delta)}$ and such that

$$ \|\nabla_x \zeta_\delta\|_{L^\infty(\mathbb{R} \times \mathbb{R}^N)} \leq \frac{C}{R(\delta)} \qquad (3.32) $$
for some $C > 0$ independent of $\delta$. Let us now define

$$h_{A,\delta} := \zeta_{\delta}(\rho_{r(\delta)} * g_{B,\delta} + 2L_{\varepsilon}r(\delta)) \in C^{\infty}(\mathbb{R} \times \mathbb{R}^{N}),$$

where $\rho$ is defined in (3.19).

* **Claim:** For $\delta > 0$ small enough so that

$$R(\delta) \leq \min\{R_*/2, R_{**}\}, \qquad (3.33)$$

$$r(\delta) \leq \frac{\kappa}{4L_{\varepsilon}}, \qquad (3.34)$$

we have supp $h_{A,\delta} \cap K \subset K \cap (K_{B,\delta} + \mathbb{B}_{R(\delta)}) \subset U_{A,\delta}$ and it holds that

$$g_{B,\delta}(p) < h_{A,\delta}(p) < g_{A,\delta}(p) \quad \text{for all } p \in U_{A,\delta}, \qquad (3.35)$$

and

$$\|h_{A,\delta}\|_{L^{\infty}(\mathbb{R}\times\mathbb{R}^N)} \lesssim r(\delta)(1+r(\delta)), \qquad (3.36)$$

$$\|\nabla_x h_{A,\delta}\|_{L^{\infty}(U)} \lesssim R(\delta)(1 + R(\delta) + r(\delta)).$$

Indeed, let $\delta$ be as above. As a preliminary, note that from (3.31) and (3.33) follow the immediate consequences

$$K \cap (K_{B,\delta} + \mathbb{B}_{R(\delta)}) \subset U_{A,\delta}, \qquad (3.37)$$

$$U + \mathbb{B}_{2R(\delta)}2R(\delta)) \subset U_{*}. \qquad (3.38)$$

The first part of our claim is thus proven thanks to (3.37). Next we show the two claims (3.35) and (3.36) sequentially.

**Proof of inequalities** (3.35) Let $p \in U_{A,\delta}$, for the left inequality, since $\rho_{r(\delta)}$ is supported in $\mathbb{B}_{r(\delta)}$ and $g_{B,\delta}$ is $L_{\varepsilon}$-Lipschitz, we find

$$\begin{aligned} h_{A,\delta}(p) - g_{B,\delta}(p) &= \zeta_{\delta}(p) \int_{\mathbb{R}^{N+1}} (g_{B,\delta}(q) - g_{B,\delta}(p) + 2L_{\varepsilon}r(\delta))\rho_{r(\delta)}(p-q) \, dq \\ &\quad - (1 - \zeta_{\delta}(p))g_{B,\delta}(p) \\ &\geq L_{\varepsilon}r(\delta)\zeta_{\delta}(p) \int_{p+\mathbb{B}_{r(\delta)}} \rho_{r(\delta)}(p-q) \, dq \\ &\quad - (1 - \zeta_{\delta}(p))g_{B,\delta}(p) \\ &= L_{\varepsilon}r(\delta)\zeta_{\delta}(p) - (1 - \zeta_{\delta}(p))g_{B,\delta}(p). \end{aligned} \qquad (3.39)$$

Now if $p \in K_{B,\delta}$, then $\zeta_{\delta}(p) = 1$ thus from (3.39) $h_{A,\delta}(p) - g_{B,\delta}(p) \geq L_{\varepsilon}r(\delta) > 0$. Otherwise if $p \notin K_{B,\delta}$, then $-g_{B,\delta}(p) > 0$ and $L_{\varepsilon}r(\delta) > 0$ thus from (3.39) again we obtain $h_{A,\delta}(p) - g_{B,\delta}(p) > 0$.
For the right inequality of (3.35) observe that

$$h_{A,\delta}(p) - g_{A,\delta}(p) = \zeta_{\delta}(p) \int_{p+\mathbb{B}_{r(\delta)}} (g_{B,\delta}(q) - g_{A,\delta}(p) + 2L_{\varepsilon}r(\delta)) \rho_{r(\delta)}(p-q) \, dq \\ - (1 - \zeta_{\delta}(p))g_{A,\delta}(p).$$

Note that for $q \in p + \mathbb{B}_{r(\delta)}$, since $g_{B,\delta}$ is $L_{\varepsilon}$-Lipschitz and using (3.26), we find

$$g_{B,\delta}(q) \leq g_{B,\delta}(p) + L_{\varepsilon}r(\delta) \\ \leq g_{A,\delta}(p) - \kappa + L_{\varepsilon}r(\delta).$$

Thus using (3.34) and the fact that $g_{A,\delta}(p) > 0$, it holds that

$$h_{A,\delta}(p) - g_{A,\delta}(p) \leq \zeta_{\delta}(p)(-\kappa + 3L_{\varepsilon}r(\delta)) - (1 - \zeta_{\delta}(p))g_{A,\delta}(p) \\ \leq -\zeta_{\delta}(p)\frac{\kappa}{4} - (1 - \zeta_{\delta}(p))g_{A,\delta}(p) \\ = -\left(\zeta_{\delta}(p)\frac{\kappa}{4} + (1 - \zeta_{\delta}(p))g_{A,\delta}(p)\right) \\ < 0.$$

□

**Proof of the bounds (3.36)** For the estimates in (3.36) it suffices to evaluate both suprema in the support of $\zeta_{\delta}$ which is contained in $K_{B,\delta} + \mathbb{B}_{N+1}(0, R(\delta))$. We look at the first estimate in (3.36). Let $p \in K_{B,\delta} + \mathbb{B}_{R(\delta)}$, we have from (3.38) that $p + \mathbb{B}_{r(\delta)} \subset \mathbb{B}_{N+1}(p, R(\delta)) \subset K_{B,\delta} + \mathbb{B}_{2R(\delta)}2R(\delta)) \subset U_*$. Hence from definition (3.27) of $M_B(\delta)$, we obtain

$$|h_{A,\delta}(p)| \leq \int_{\mathbb{R}^{N+1}} g_{B,\delta}(q) \rho_{r(\delta)}(q-p) \, dq + 2L_{\varepsilon}r(\delta) \\ \leq M_B(\delta) + 2L_{\varepsilon}r(\delta) \\ = r(\delta)(2L_{\varepsilon} + r(\delta)).$$

For the second estimate in (3.36), again since $p + \mathbb{B}_{r(\delta)} \subset U_*$, we have recalling (3.32) and the definitions (3.28)-(3.29) of $r(\delta)$ and $R(\delta)$,

$$|\nabla_x h_{A,\delta}(p)| \leq |\nabla_x \zeta_{\delta}(p)| (|\rho_{r(\delta)} * g_{B,\delta}(p)| + 2L_{\varepsilon}r(\delta)) + |\nabla_x [\rho_{r(\delta)}] * g_{B,\delta}(p)| \\ \leq C \frac{M_B(\delta) + 2L_{\varepsilon}r(\delta)}{R(\delta)} + \frac{1}{r(\delta)} \|\nabla_x \rho\|_{L^1} M_B(\delta) \\ \leq C R(\delta)(2L_{\varepsilon} + \|\nabla_x \rho\|_{L^1} R(\delta) + r(\delta)).$$

□
A symmetric construction of a function $-h_{B,\delta}$ holds, where from the equation (3.30) on, $g_{A,\delta}$ is replaced by $-g_{B,\delta}$ and $g_{B,\delta}$ is replaced by $-g_{A,\delta}$. This function is such that that supp $h_{B,\delta} \subset K \cap (K_{A,\delta} + \mathbb{B}_{R(\delta)}) \subset U_{B,\delta}$,

$$ g_{B,\delta}(q) < h_{B,\delta}(q) < g_{A,\delta}(q) \quad \text{for all } q \in U_{B,\delta}, \qquad (3.40) $$

and

$$ \begin{aligned} \|h_{B,\delta}\|_{L^{\infty}(\mathbb{R}\times\mathbb{R}^N)} &\lesssim r(\delta)(1+r(\delta)) \\ \|\nabla_x h_{B,\delta}\|_{L^{\infty}(U)} &\lesssim R(\delta)(1+R(\delta)+r(\delta)) \end{aligned} \qquad (3.41) $$

$$ \text{where } U_{B,\delta} := \{p \in K \mid g_{B,\delta} < 0\}, \quad K_{A,\delta} := \{p \in K \mid g_{A,\delta} \le 0\}. $$

Now since $K_{B,\delta}$ and $K_{A,\delta}$ are compact and disjoint from (3.26), for small enough $\delta > 0$, $h_{A,\delta}$ and $h_{B,\delta}$ have disjoint supports in $\bar{U}$. Remark further that $U \setminus \bar{O}' \subset K = U_{A,\delta} \cup U_{B,\delta} = \bar{U} \setminus \bar{O}'$. Letting

$$ h_{\delta} := h_{A,\delta} + h_{B,\delta}, $$

it is now straightforward to see that for $\delta > 0$ small enough (3.24) follows from (3.35)-(3.40) and that (3.25) follows from (3.36)-(3.41), which concludes our proof. $\square$

# 4 Proof of Theorem 1.6

The modified interposition is constructed as the junction of a cylindrical extension of a $C^{1,1}$ interposition between $A(0)$ and $B(0)$ and a piecewise $C^{1,1}$ $\varepsilon$-interposition between $A$ and $B$ given by Theorem 1.3.

Let us introduce some notations for convenience: $D := \text{dist}^0(A, B)$, $D_\kappa := \text{dist}^0(A_\kappa, B_\kappa)$ and $D_0 := \text{dist}(A(0), B(0))$. The assumptions of Theorem 1.6 translate into the following inequalities,

$$ D < D_\kappa < D_0. \qquad (4.1) $$

* Since $A(0), B(0) \subset \mathbb{R}^N$ are closed and disjoint with $A(0)$ compact, from Corollary 2.2 there exists a $d_N$-interposition $\omega$ with $C^{1,1}$ such that $\lambda := \frac{\text{dist}(\partial\omega, A(0))}{D_0} \in (0, 1)$ satisfies

$$ \frac{D_\kappa + \frac{h}{4}}{2D_\kappa + h} \le \lambda \le \frac{D_\kappa + \frac{3h}{4}}{2D_\kappa + h}, \qquad (4.2) $$

where $h := \frac{D_0 - D_\kappa}{2} > 0$ from (4.1).

* Let $x \in \partial\omega$, denote $p_x := (0, x)$ and observe that, since $A$ and $B$ are closed in $\mathbb{R} \times \mathbb{R}^N$,

$$ d^\varepsilon(p_x, A_\kappa) \uparrow_{\varepsilon \to 0} d_N(x, A(0)), $$

$$ d^\varepsilon(p_x, B_\kappa) \uparrow_{\varepsilon \to 0} d_N(x, B(0)). $$
Consequently, there exists $\varepsilon_x(A_\kappa, B_\kappa) > 0$ such that for each $0 < \varepsilon \le \varepsilon_x$, since $D_\kappa < D_0$ and using the definition of $\lambda$,

$$ 
\begin{aligned}
d^{\varepsilon}(p_x, A_\kappa) &\ge \lambda(D_\kappa + h), \\
d^{\varepsilon}(p_x, B_\kappa) &\ge (1 - \lambda)(D_\kappa + h).
\end{aligned}
 $$

• Since the maps $p \mapsto d^{\varepsilon_x}(p, A_\kappa/B_\kappa)$ are $\frac{1}{\varepsilon_x}$-Lipschitz continuous functions, there exists for each $p_x$ a neighborhood $U(p_x) \subset \mathbb{R} \times \mathbb{R}^N$ such that for all $p \in U(p_x)$,

$$ 
\begin{aligned}
d^{\varepsilon_x}(p, A_\kappa) &\ge \lambda\left(D_\kappa + \frac{h}{2}\right), \\
d^{\varepsilon_x}(p, B_\kappa) &\ge (1 - \lambda)\left(D_\kappa + \frac{h}{2}\right).
\end{aligned}
 $$

The family of open sets $\{U(q_x)\}_{x \in \partial\omega}$ covers $\{0\} \times \partial\omega$, by compactness there exists a finite family of points $\{x_n\}_{1 \le n \le N} \subset \partial\omega$ such that

$$ \{0\} \times \partial\omega \subset U := \bigcup_{n=1}^{N} U(q_{x_n}). $$

Now $\varepsilon \mapsto d^{\varepsilon}(\cdot, A_\kappa/B_\kappa)$ is non-decreasing, thus we set $\bar{\varepsilon}_*(A_\kappa, B_\kappa) := \min_{1 \le n \le N} \{\varepsilon_{x_n}\} > 0$ so that for all $0 < \varepsilon \le \bar{\varepsilon}_*$, and for all $p \in U$,

$$ 
\begin{aligned}
d^{\varepsilon}(p, A_\kappa) &\ge \lambda\left(D_\kappa + \frac{h}{2}\right), \\
d^{\varepsilon}(p, B_\kappa) &\ge (1 - \lambda)\left(D_\kappa + \frac{h}{2}\right).
\end{aligned}
 $$

Since $U$ is an open neighborhood of the compact set $\{0\} \times \partial\omega$, $B^c$ is an open neighborhood of $\{0\} \times \omega$ and $A^c$ is an open neighborhood of $\{0\} \times \overline{\omega^c}$ with $A$ compact, there exists $\delta \in (0, \frac{\kappa}{2})$ such that $[-\delta, \delta] \times \partial\omega \subset U$ and

$$ A_{\delta} \subset \subset [-\delta, \delta] \times \omega \subset \subset B_{\delta}^{c}. \qquad (4.4) $$

• Let us set $\varepsilon_* := \min\{\bar{\varepsilon}_*, \frac{\kappa}{2D_\kappa+h}\} > 0$. Let $\varepsilon \in (0, \varepsilon_2^*)$ and let $p = (t, x) \in [-\delta, \delta] \times \partial\omega$, we will show that we in fact have

$$ d^{\varepsilon}(p, A) \ge \lambda\left(D_{\kappa} + \frac{h}{2}\right), \qquad (4.5) $$

$$ d^{\varepsilon}(p, B) \ge (1 - \lambda)\left(D_{\kappa} + \frac{h}{2}\right). \qquad (4.6) $$

We only show the argument for (4.5), the argument for (4.6) is analogous. Let $p_A = (t_A, x_A) \in A$ be a point realizing the distance, i.e. such that
$d^{\epsilon}(p, A) = \|p - p_A\|^{\epsilon}$. If $t_A \in [-\kappa, \kappa]$, then $d^{\epsilon}(p, A) = d^{\epsilon}(p, A_{\kappa})$ and (4.3) concludes. Otherwise, then $t_A \notin [-\kappa, \kappa]$, in particular because $\delta \le \kappa/2$, $|t - t_A| \ge \kappa/2$ and thus from the definition of $\epsilon_*$ and since $\lambda \in (0, 1)$,

$$ 
\begin{aligned}
d^{\epsilon}(p, A) &\ge \frac{|t - t_A|}{\epsilon} \\
&\ge \frac{\kappa}{2\epsilon} \\
&\ge D_{\kappa} + \frac{h}{2} \\
&\ge \lambda \left( D_{\kappa} + \frac{h}{2} \right),
\end{aligned}
 $$

which concludes our argument.

* Let $\epsilon \in (0, \min\{\epsilon_*(A_\kappa, B_\kappa), \epsilon_\Gamma(A, B)\})$. From Theorem 1.5, there exists a non-characteristic $\epsilon$-interposition $K_\epsilon$ between A and B and by density we can choose $\sigma := \text{dist}^\epsilon(\partial K_\epsilon, A) > 0$ such that

$$ \left| \sigma - \frac{1}{2} \text{dist}^{\epsilon}(A, B) \right| \le \frac{h}{8}. \quad (4.7) $$

We find from (4.5), the definition (4.2) of $\lambda$ and the fact that $D_\kappa > D \ge \text{dist}^\epsilon(A, B)$, that for all $p \in [-\delta, \delta] \times \partial\omega$,

$$ 
\begin{aligned}
d^{\epsilon}(p, A) &\ge \lambda(D_{\kappa} + \frac{h}{2}) \\
&\ge \frac{1}{2}D_{\kappa} + \frac{h}{8} \\
&> \frac{1}{2}\text{dist}^{\epsilon}(A, B) + \frac{h}{8}
\end{aligned}
 $$

Using (4.7), we obtain the strict inequality

$$ d^{\epsilon}(p, A) > \sigma. \quad (4.8) $$

In a very similar manner, we obtain for all $p \in [-\delta, \delta] \times \partial\omega$, the corresponding strict inequality for B,

$$ d^{\epsilon}(p, B) > \text{dist}^{\epsilon}(A, B) - \sigma. \quad (4.9) $$

* Let us now will apply Corollary 2.5 with the following variables

$$ K := K_\epsilon, $$

$$ \mathcal{L} := [-\delta, \delta] \times \omega, $$

$$ S := [-\delta/2, \delta/2] \times \mathbb{R}^N. $$

The hypotheses are checked directly using (4.4), (4.8) and (4.9). As a consequence, the junction

$$ \tilde{K}_{\epsilon} := K_{\epsilon}|_{[-\delta/2, \delta/2]^c} \cup [-\delta/2, \delta/2] \times \omega $$
is a $\varepsilon$-interposition between $A$ and $B$. The fact that $\tilde{K}_\varepsilon$ is $C_{t,pw}^{1,1}$ follows directly from the fact that $K_\varepsilon \in C_{t,pw}^{1,1}(\mathbb{R})$ and that $[-\delta/2, \delta/2] \times \omega \in C_t^{1,1}([-\delta/2, \delta/2])$. Finally $K_\varepsilon$ is non-characteristic because the inequalities in (4.8)-(4.9) are strict, thus any critical point $p \in \partial \tilde{K}_\varepsilon$ must in fact be a non-characteristic point of $\partial K_\varepsilon$.

□

## Acknowledgments

The author would like to thank Pierre Cardaliaguet for an insightful discussion as well as Boris Andreianov for interesting exchanges that lead to significant improvements of this work.

This work is partially supported by the Deutsche Forschungsgemeinschaft DFG through the cluster of excellence 'SimTech' under EXC 2075-390740016.
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