# The medial axis of any closed bounded set is locally Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms

Hana Dal Poz Kouřimská<fnref n="1" /> · André Lieutier<fnref n="2" /> · Mathijs Wintraecken<fnref n="3" />

Received: 13 October 2023 / Revised: 22 January 2025 / Accepted: 6 June 2025
© The Author(s) 2025

## Abstract

We prove that the medial axis of a closed set is locally Hausdorff stable in the following sense: Let $S \subseteq \mathbb{R}^d$ be a fixed compact set and $S(c, r)$ some sphere with radius $r$ containing $S$ in its interior. Consider the space of $C^{1,1}$ diffeomorphisms of $\mathbb{R}^d$ to itself, which keep the exterior of $S(c, r)$ invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of $\mathbb{R}^d$ (endowed with the Hausdorff distance), mapping a diffeomorphism $F$ to the closure of the medial axis of $F(S \cup S(c, r))$, is Lipschitz. A similar statement holds if $S$ is non-compact but the Hausdorff distance between $S$ and $\mathbb{R}^d$ is bounded; in other words, every point in $\mathbb{R}^d$ has a point in $S$ nearby. The latter statement can further be localized at the cost of having to consider two one-sided Hausdorff distances. Our result extends result of Chazal and Soufflet on the stability of the medial axis of $C^2$ manifolds under $C^2$ ambient diffeomorphisms.

**Keywords** Medial axis · Hausdorff distance · Lipschitz continuity

**Mathematics Subject Classification** 65D18 · 54H30

<figure><img src="image_1.png" /></figure>

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

<footnote>Hana Dal Poz Kouřimská
hana.dal.poz.kourimska@uni-potsdam.de</footnote>

<footnote>Mathijs Wintraecken
mathijs.wintraecken@inria.fr</footnote>

<footnote>André Lieutier
andre.lieutier@gmail.com</footnote>

<footnote n="1">¹ Angewandte Geometrie & Topologie, Institut für Mathematik, University of Potsdam, Campus Golm, Haus 9, Karl-Liebknecht-Str. 24-25, Potsdam 14476, Germany</footnote>

<footnote n="2">² No affiliation, Aix-en-Provence, France</footnote>

<footnote n="3">³ Centre Inria d'Université Côte d'Azur, Université Côte d'Azur, Inria, 2004 Rte des Lucioles, Valbonne 06902, France</footnote>
# 1 Introduction

In Federer (1959), Federer introduced the *reach* of a (closed) set $S \subset \mathbb{R}^d$ as the infimum over all points in $S$ of the distance from these points to the *medial axis* $ax(S)$, the set of points in $\mathbb{R}^d$ for which the closest point in $S$ is not unique. Federer also introduced the *reach* at a point $p \in S$ to be the distance from $p$ to the *medial axis* of $S$. We now call this quantity the *local feature size* (Amenta and Bern 1999) and denote it by $\text{lfs}(p)$.

Federer proved that the *reach* is stable under $C^{1,1}$ diffeomorphisms of the ambient space. Here, a $C^{1,1}$ map is a $C^1$ map whose derivative is Lipschitz, and a $C^{1,1}$ diffeomorphism is a $C^{1,1}$ bijective map whose inverse is also $C^{1,1}$. Chazal and Soufflet (2004) proved that the *medial axis* is stable with respect to the Hausdorff distance under ambient diffeomorphisms, but under stronger assumptions than the work of Federer, namely assuming that $S$ is a $C^2$ manifold and the distortion is a $C^2$ diffeomorphism of the ambient space. Chazal and Soufflet based their work on earlier results by Blaschke (1916), which were not as strong as Federer's.

In this paper we extend the stability result of the *medial axis*. More concretely, we generalize the result of Chazal and Soufflet (2004) to arbitrary closed sets and $C^{1,1}$ diffeomorphisms of the ambient space; we show that the Hausdorff distance between the two medial axes of the closed set and its image is bounded in terms of Lipschitz constants stemming from the diffeomorphism of the ambient space. Let $S$ be a compact and therefore bounded subset of $\mathbb{R}^d$. Suppose that $S(c, r) \subseteq \mathbb{R}^d$ is a sphere that contains $S$ in its interior, and $F$ is a diffeomorphism of $\mathbb{R}^d$ to itself that leaves the exterior of $S(c, r)$ invariant. Denote the Hausdorff distance by $d_H$. We show (see Theorem 17) that

$$d_H(\mathrm{ax}(S \cup S(c, r)), \mathrm{ax}(F(S \cup S(c, r)))) = \mathcal{O}(r^2 \varepsilon).$$

Here, $r$ is the radius of $S(c, r)$ and $\varepsilon$ bounds how close the diffeomorphism $F$ is to the identity.

A similar result holds for $d_H(\mathrm{ax}(S), \mathrm{ax}(F(S)))$ if $S$ is non-compact, does not contain a sphere $S(c, r)$, but instead $d_H(S, \mathbb{R}^d) \le r$. In the latter case we do not require the diffeomorphism $F$ to leave the exterior of some sphere $S(c, r)$ invariant (but $F$ still needs to be close to the identity). This result can be localized at the cost of having to consider two one-sided Hausdorff distances. We also localize this result at the cost of considering two one-sided Hausdorff distances. Our result relies on the work of Federer (1959) and in fact shortens the proof in Chazal and Soufflet (2004) significantly.

Our bounds on the Hausdorff distance say nothing about the topology of the *medial axis*, which is known to be highly unstable (see e.g. Attali et al. (2009) and Fig. 1), although it preserves the homotopy type (see Lieutier (2004)).

**Contribution and related work** Our work differs from the majority of the literature in three essential ways:

**Firstly, we make no assumptions on the set we consider apart from that it is closed and either bounded or dense.** The stability of the *medial axis* of (piecewise)
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Small (in terms of the Hausdorff distance) bumps on an even smooth curve or surface can create large new branches of the medial axis. However, these bumps in are not small in the $C^2$ or $C^{1,1}$ sense (roughly speaking the second order derivatives will be large) and therefore cannot be the consequence of an ambient diffeomorphism whose distortion is small in a $C^2$ or $C^{1,1}$ sense</figcaption></figure>

smooth manifolds has been the object of intense study, see for example (Chazal and Soufflet 2004; Mather 1983; Thom 1972; van Manen 2007; Wolter 1993; Damon 2021, 2008; Damon and Gasparovic 2017; Gasparovic 2012; Wall 1977). However, the manifold assumption is impossible to achieve in many applications — such as in the context of astrophysics, one of the main motivations of this paper.

**Secondly, we achieve stability without pruning the medial axis.** This contrasts with a large body of work, such as Attali and Montanvert (1997); Chazal and Lieutier (2005); Damon (2021); Lieutier and Wintraecken (2023). Not having to prune the medial axis is a significant advantage. On the downside, we limit the changes of the considered set to those induced by ambient diffeomorphisms. Nevertheless, given the standard examples of the instability of the medial axis — see for example (Attali et al. 2009) — we believe these limitations are near to the weakest assumptions necessary for Hausdorff stability. Within the context of ambient homeomorphisms, the results we obtain are close to optimal, as we specify in Remark 19.

**Thirdly, our results hold for sets in arbitrary dimensions and are not sensitive to the dimension of the set itself.** A large part of the related work only investigates sets of low dimensions or codimension one manifolds, although there are some notable exceptions such as Wall (1977), see also Damon and Gasparovic (2017), and Chazal and Lieutier (2005); Lieutier and Wintraecken (2023).

## Motivation

The medial axis has many real world applications — among others, in robot motion planning (Latombe 2012), triangulation algorithms (Amenta et al. 2001), graphics (Tagliasacchi et al. 2016), vision (Hung et al. 2012; Lescroart and Biederman 2013), and shape recognition, segmentation, and learning (Chambers et al. 2018; Ho and Dyer 1986; Shaked and Bruckstein 1998; Demir et al. 2019; Yan et al. 2016; Trinh and Kimia 2011; Hu et al. 2019; Rezanejad et al. 2019). See also the overviews (Tagliasacchi et al. 2016; Saha et al. 2016). The reach — the distance between a set and its medial axis — is a central concept in manifold learning (Aamari and Knop 2021; Aamari and Levrard 2018; Fefferman et al. 2018, 2019, 2020; Sober and Levin 2020).

The motivation of this paper is twofold: Firstly, we tackle the following challenge from the processing of images collected with optical devices which use lenses — such as cameras or telescopes: A shape extracted from such an image may be imprecise due to the imperfection of the lenses.

Our result implies that the medial axis of such a shape is stable under these imperfections. As a consequence, the outcome of any shape recognition or shape segmentation algorithm based on the medial axis will be stable.
In addition to the disciplines listed at the beginning of this paragraph, the stability of the medial axis is desirable in astrophysics, in particular for shape analysis and automated shape identification in observational astronomy. Observational astronomers are interested in reconstructing objects like stars or galaxies, and their place in the universe from data gathered by telescopes. They can deduce the distance from the object to the observer thanks to so-called standard candles or red shift (Fernie 1969; Peebles 1993; Coble et al. 2020). However, the image gets distorted due to optical effects — either through gravitational lensing ((Bartelmann 2010)) or lensing inside the telescope itself ((Tang et al. 2017)).

Such a distortion can be modeled as a diffeomorphism of the ambient space. At the same time, this problem cannot be tackled using the result by Chazal and Soufflet (2004), since the observed objects might not be smooth — for example due to interactions with shock waves or jets. In addition, with our method astrophysicists can not only reconstruct objects in space (3D), but also in spacetime (4D).

Another context where the removal of the assumption that the set is a (smooth) manifold is important is biology, as branching structures are ubiquitous in nature. In fact, it was questions from biology that motivated the ‘introduction’<fnref n="1" /> of the medial axis by Blum (1967).

The second motivation is more formal in nature: The stability of the medial axis is instrumental in establishing its computability. Indeed, when proving properties of algorithms based on the medial axis, authors generally assume the real RAM model.<fnref n="2" /> However, as was recently argued in Lieutier and Wintraecken (2023), the medial axis needs to be stable in order to be computable in more realistic models of computation.

There is a more practical component to this formal question: It is not a priori clear if using possibly noisy real world data or the output of other computer programs as input for these algorithms yields answers that are close to the ground truth. To be able to prove that the output is correct, we need (numerical) stability of the medial axis.

**Outline** After revisiting preliminaries and known results in Section 2, we state the main stability result in Section 3. In Section 4 we reformulate this result in terms of norms on Banach spaces. This also exhibits the fact that the stability of the medial axis is Lipschitz in the following sense: We think of the set $\mathcal{S}$ as fixed and consider the map from the space of diffeomorphisms (endowed with a norm which makes it a Banach space) to the space of closed subsets of $\mathbb{R}^d$ (endowed with the Hausdorff distance), mapping each diffeomorphism $F: \mathbb{R}^d \to \mathbb{R}^d$ that leaves the exterior of $S(c, r)$ invariant to the closure of the medial axis of $F(\mathcal{S} \cup S(c, r))$. In Section 5, we replace the assumption on $F$ to leave the exterior of $S(c, r)$ invariant by several alternatives, that imply the stability of the medial axis in one form or other. We conclude with some future work.

<footnote n="1">¹ The medial axis was studied before by Erdős (1945); Erdős (1946) in a different context.</footnote>

<footnote n="2">² The real RAM model is a standard, albeit non-realistic, assumption in Computational Geometry. It assumes one can calculate precisely with real numbers, instead of using 0s and 1s (which is the usual assumption in computer science).</footnote>
## 2 Preliminaries: Sets of positive reach and the closest point projection

In this section we recall some definitions and results concerning the medial axis and sets of positive reach. Essentially, we need three ingredients from the literature to prove our main theorem: the notions related to the closest point projection, the properties of the generalized normal and tangent spaces, and Federer's result on the stability of the reach under ambient diffeomorphisms (Federer 1959).

We write $d(\cdot, \cdot)$ for the Euclidean distance between two points, and the distance between a point and a set. That is, for any closed set $S$ and point $p$,

$$d(p, S) = \inf_{q \in S} d(p, q).$$

We denote the Hausdorff distance between two sets $A, B \subseteq \mathbb{R}^d$ by $d_H(A, B)$:

$$d_H(A, B) = \max \left( \sup_{a \in A} d(a, B), \sup_{b \in B} d(b, A) \right).$$

We recall that the one-sided Hausdorff distance from $A$ to $B$ is defined as:

$$d_H^o(A, B) = \sup_{a \in A} d(a, B).$$

We write $B(c, r)$, resp. $S(c, r)$, to denote balls, resp. spheres, with centre $c$ and radius $r$. If no centre is specified, the centre is the origin $0$. Lastly, $|\cdot|$ denotes the Euclidean norm, and $||\cdot||$ the operator norm induced by the Euclidean norm.

**The closest point projection and related notions** The projection of points in the ambient space $\mathbb{R}^d$ to the (set of) closest point(s) of the set $S \subseteq \mathbb{R}^d$ is denoted by $\pi_S$, and illustrated in Figure 2.

The *medial axis* of $S$ is the set of all points $p \in \mathbb{R}^d$ where the set $\pi_S(p)$ consists of more than one point:

$$\mathrm{ax}(S) = \{ p \in \mathbb{R}^d \mid \#\pi_S(p) > 1 \}.$$

Here, $\#\pi_S(p)$ denotes the cardinality of the set $\pi_S(p)$.

For a point $p \in S$, the *local feature size of* $p$ is the distance from $p$ to the medial axis of the set $S$:

$$\mathrm{lfs}(p) = d(p, \mathrm{ax}(S)).$$

Finally, the *reach* of the set $S$ is the infimum of the local feature size over all its points:

$$\mathrm{rch}(S) = \inf_{p \in S} \mathrm{lfs}(p) = \inf_{p \in S} d(p, \mathrm{ax}(S)).$$
<figure><img src="image_4.png" /><figcaption>**Fig. 2** The closest point projection to the set $\mathcal{S}$ of four points in $\mathbb{R}^2$. When a point lies on the medial axis $\text{ax}(\mathcal{S})$, the closest point projection consists of more points</figcaption></figure>

Throughout this paper we assume that $\mathcal{S} \subseteq \mathbb{R}^d$ is a closed set. We often assume that the set $\mathcal{S}$ is compact and contains its own bounding sphere. To be more concrete: Given a compact set $\mathcal{S}'$, we let $B$ be a closed ball containing $\mathcal{S}'$. We then define $\mathcal{S} = \mathcal{S}' \cup \partial B = \mathcal{S}' \cup \mathcal{S}(c, r)$ and say that $\mathcal{S}$ is *sphere bounded*. We frequently assume without loss of generality that the centre $c$ of $\mathcal{S}(c, r)$ is the origin $0$, in which case we drop $c$ from the notation.

The assumption that $\mathcal{S}$ is sphere bounded localizes the problem, in particular it ensures that the medial axis is bounded. We will see in Section 5 that there are alternative ways to localize the problem; we can for example assume that $d_H(\mathcal{S}, \mathbb{R}^d) \le r$ and a similar assumption for $F(\mathcal{S})$. The latter assumptions can be replaced by more local conditions, but this unfortunately comes at the cost of having to consider two one-sided Hausdorff distances.

Some localization, such as the addition of the bounding sphere $\partial B$ to the set $\mathcal{S}'$, is necessary to obtain a bound on the Hausdorff distance between the two medial axes of the set $\mathcal{S}$ and its image under the ambient diffeomorphism.

Indeed, consider the following example, illustrated in Figure 3. Let the set $\mathcal{S}$ consist of two points in the plane, $\mathcal{S} = \{p, q\} \subseteq \mathbb{R}^2$. The medial axis of $\mathcal{S}$ is then the bisector line of $p$ and $q$. After a generic perturbation $F$ of $p$ and $q$ — that is, not a translation and not a perturbation in the direction $\pm(p - q)$ — the bisector line $\text{ax}(F(\mathcal{S}))$ of the perturbed points intersects the bisector $\text{ax}(\mathcal{S})$ of the original pair. The Hausdorff distance between these two non-parallel lines is infinite, and thus unboundable. If, however, we restrict ourselves to a ball around the origin of size $r$, the Hausdorff distance between the two restricted medial axes is of order $O(r\theta)$, where $\theta$ is the angle between the vectors $p - q$ and $F(p) - F(q)$.

A recurring strategy in this article is to start at a point $p$ on the set $\mathcal{S}$, move away from this point in a 'normal' direction, and see if by projecting using the closest point projection $\pi_{\mathcal{S}}$ we get back to $p$. To this end, we define the *projection range*.
<figure><img src="image_5.png" /><figcaption>Fig. 3 In black the set $S$ and its medial axis, in light blue the perturbed set and its medial axis. Since the lines $ax(S)$ and $ax(F(S))$ are non-parallel, the Hausdorff distance between them is infinite. Hence it is impossible to give a bound on the distance between the medial axes without localizing</figcaption></figure>

**Definition 1** (Projection range) Let $p \in S$ be a point and $v \in \mathbb{R}^d$ a vector. The projection range $d(p, v, \pi_S)$ in direction $v$ is the maximal distance one can travel from $p$ along $v$ such that the closest point projection yields only the point $p$:

$$d(p, v, \pi_S) = \sup\{\lambda \in \mathbb{R} \mid \pi_S(p + \lambda v) = \{p\}\}.$$

Since $\pi_S(p) = \{p\}$, the projection range is canonically non-negative. Furthermore, the directions for which the range is positive are key to our study, because of the following property:

**Lemma 2** (Theorem 4.8 (6) of Federer (1959)) Consider a point $p \in S$ and a vector $v \in \mathbb{R}^d$. If

$$0 < d(p, v, \pi_S) < \infty,$$

then $p + d(p, v, \pi_S) \cdot v \in \overline{ax(S)}$, where $\overline{X}$ denotes the closure of $X$.

We call these special directions $v$ back projection vectors:

**Definition 3** (Unit back projection vectors)

For a point $p \in S$, UBP($p, S$) is the set of unit vectors with a positive projection range:

$$UBP(p, S) = \{u \in \mathbb{R}^d \mid |u| = 1 \text{ and } 0 < d(p, u, \pi_S) < \infty\}.$$

We further define

$$UBP(S) = \{ (p, u) \in S \times \mathbb{R}^d \mid u \in UBP(p, S) \},$$

$$BP(S) = \{ (p, \lambda u) \in S \times \mathbb{R}^d \mid (p, u) \in UBP(S), \lambda \ge 0 \}.$$

These definitions are illustrated in Fig. 4

Thanks to Lemma 2, the following map is well-defined:

$$\pi_{\text{ax},S}: \text{UBP}(S) \to \overline{ax(S)}, \quad (p, u) \mapsto p + d(p, u, \pi_S)u. \quad (1)$$
<figure><img src="image_6.png" /><figcaption>**Fig. 4** The projection range $d(p_1, w, \pi_S)$ equals infinity if $w \in \text{UBP}(p_1, S)$ and zero otherwise. For $p_2, d(p_2, -v, \pi_S)$ is infinite, $d(p_2, v, \pi_S)$ is finite and non-zero, and $d(p_2, w, \pi_S) = 0$ for $w \neq \pm v$. Hence, the set of unit projection vectors equals $\text{UBP}(p_2, S) = \{v, -v\}$</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5** The (affine) generalized tangent and normal spaces of four points in the set $S \subset \mathbb{R}^2$, in light blue and violet, respectively</figcaption></figure>

**The generalized tangent and normal space** Back projection vectors are intricately related to the generalized tangent and normal spaces.

**Definition 4** (Definitions 4.3 and 4.4 of Federer (1959)) Let $p \in S$. The generalized tangent space $\text{Tan}(p, S)$ is the set of vectors $u \in \mathbb{R}^d$, such that either $u = 0$ or, for every $\varepsilon > 0$ there exists a point $q \in S$ with

$$0 < |q - p| < \varepsilon \quad \text{and} \quad \left| \frac{q - p}{|q - p|} - \frac{u}{|u|} \right| < \varepsilon.$$

The generalized normal space $\text{Nor}(p, S)$ consists of vectors $v \in \mathbb{R}^d$ such that $\langle v, u \rangle \le 0$ for all $u \in \text{Tan}(p, S)$. Vectors contained in the generalized tangent, resp. normal, space are called *tangent*, resp. *normal*, to $S$ at $p$.

The generalized tangent and normal spaces are illustrated in Figure 5.

**Stability of the reach under ambient diffeomorphisms**

Our last ingredient is the following result by Federer. We denote derivative by $D$.
**Theorem 5** (Stability of the reach under ambient diffeomorphisms, Theorem 4.19 of Federer (1959)) Pick two constants $0 < t < rch(S)$ and $s > 0$. If the map

$$F : \{x \in \mathbb{R}^d \mid d(x, S) < s\} \to \mathbb{R}^n$$

is injective and continuously differentiable, and the maps $F, F^{-1}$, and $DF$ are Lipschitz continuous with Lipschitz constants $Lip(F), Lip(F^{-1}), Lip(DF)$, respectively, then the reach $rch(F(S))$ of the image of the set $S$ under the map $F$ is lower-bounded by

$$rch(F(S)) \geq \min \left\{ \frac{s}{Lip(F^{-1})}, \frac{1}{\left( \frac{Lip(F)}{t} + Lip(DF) \right) (Lip(F^{-1}))^2} \right\}.$$

**3 Stability of the medial axis under ambient diffeomorphisms**

In this section we present the main result of this paper, Theorem 16. This theorem extends earlier work by Chazal and Soufflet (2004). Its proof relies on Federer's result (Federer 1959) on the stability of the reach, Theorem 5. It was a surprise to the authors that no assumption on the set (apart from closedness) needed to be made, and that the techniques used were that simple and well established; they go back to Federer. In fact, the authors at first envisioned a far more elaborate argument assuming the set had positive $\mu$-reach (Chazal et al. 2009).

To give a more geometrical interpretation to some of the results by Federer we introduce the concept of a weakly tangent sphere and ball, and a maximal empty weakly tangent ball.

**Definition 6** (Weakly tangent sphere and ball) Let $p \in S$. A sphere is called weakly tangent to $S$ at $p$ if it contains the point $p$ and its centre lies in the (translated) generalized normal space $Nor(p, S) + p$. In other words, spheres weakly tangent to $S$ at $p$ are spheres with centres $p + v$ and radii $|v|$, for a vector $v \in Nor(p, S)$.

A ball is called weakly tangent to $S$ at $p$ if its boundary sphere is weakly tangent to $S$ at $p$.

**Remark 7** Using the definition of $Nor(p, S)$, a weakly tangent ball can also be defined as follows: A ball $B(c, r)$ is weakly tangent at $p$ if and only if its centre $c$ and radius $r$ satisfy

$$(p + \mathrm{Tan}(S, p)) \cap B(c, r) = \{p\}.$$

The following lemma (8) essentially tells us that a family of weakly tangent balls $\{B(p + \lambda v, \lambda|v|)\}_{\lambda \ge 0}$ contains at most one which is maximal with respect to inclusion among those whose interior is disjoint from the set $S$. Two such families are illustrated in Figure 6.

**Lemma 8** Let $p \in S$ and $v \in \mathbb{R}^d$, and suppose that for some $\lambda > 0$ we have $\pi_S(p + \lambda v) \neq \{p\}$. Then, for all $\lambda' \ge \lambda$, we have $\pi_S(p + \lambda'v) \neq \{p\}$ and for all $\lambda' > \lambda$, that $p \notin \pi_S(p + \lambda'v)$.
<figure><img src="image_8.png" /><figcaption>**Fig. 6** Two families of balls weakly tangent to the set $S \subset \mathbb{R}^2$ (in blue). Each family contains a unique maximal empty ball (in purple). Notice that the centre of the maximal empty ball weakly tangent at the point $p_1$ lies at the medial axis $ax(S)$, while the centre of the maximal empty ball weakly tangent at the point $p_2$ only lies at its closure, $\overline{ax(S)}$</figcaption></figure>

**Proof** We first note that the statement is empty if $v = 0$. Let $v \neq 0$, and consider the nested family of balls $B(p + \lambda'v, \lambda'|v|)$, parametrized by $\lambda' > 0$.

Because $\pi_S(p + \lambda v) \neq p$, the (closed) ball $B(p + \lambda v, \lambda|v|)$ contains a point $q \in S$ other than $p$. Since the balls $B(p + \lambda'v, \lambda'|v|)$ are nested, the point $q$ lies inside every ball $B(p + \lambda'v, \lambda'|v|)$ with $\lambda' \ge \lambda$. Moreover, $q$ lies in the interior of $B(p + \lambda'v, \lambda'|v|)$ for $\lambda' > \lambda$. Hence, for every $\lambda' \ge \lambda$, we have that $\pi_S(p + \lambda'v) \neq \{p\}$ and for $\lambda' > \lambda$, that $p \notin \pi_S(p + \lambda'v)$. $\square$

The maximal ball in the family of weakly tangent balls $\{B(p + \lambda v, \lambda|v|)\}_{\lambda \ge 0}$ with respect to inclusion (as described in Lemma 8) is called maximal empty. For the purpose of this article, we define maximal empty balls in terms of unit back projection vectors (Definition 3). To see that each maximal empty ball is indeed weakly tangent, we emphasise:

**Lemma 9** If $(p, v) \in BP(S)$, then $(p, v) \in \text{Nor}(S)$. That is, $BP(S) \subseteq \text{Nor}(S)$. In particular, for any pair $(p, u) \in \text{UBP}(S)$ and radius $\lambda \ge 0$, the ball $B(p + \lambda u, \lambda)$ is weakly tangent to $S$.

**Remark 10** For general closed sets, the converse of Lemma 9, that is, $\text{Nor}(S) \subseteq BP(S)$, is not true. One counter-example is the graph of the function $x \mapsto |x|^{3/2}$ at the origin. However, the inclusion $\text{Nor}(S) \subseteq BP(S)$ holds for sets of positive reach thanks to (Federer (1959), Theorem 4.8 (12)), which we recall as Lemma 23 in the appendix.

**Definition 11** (Maximal empty weakly tangent ball) Let $(p, u) \in \text{UBP}(S)$. A weakly tangent ball $B(p + \lambda u, \lambda)$ is called maximal empty to $S$ if $\lambda = d(p, u, \pi_S)$, or, equivalently, if $\pi_{\text{ax},S}(p, u) = p + \lambda u$.

(Maximal empty) weakly tangent balls satisfy the following properties. Let $(p, u) \in \text{UBP}(S)$.

* For any radius $0 < \lambda \le d(p, u, \pi_S)$, the interior of the ball $B(p + \lambda u, \lambda)$ is disjoint from the set $S$. This follows directly from Definition 11 and Lemma 8.
<figure><img src="image_9.png" /><figcaption>Fig. 7 The setting of Theorem 16</figcaption></figure>

• The centres of maximal empty weakly tangent balls lie on the closure of the medial axis of $S$. This is due to Lemma 2 and the definition of the map $\pi_{\text{ax},S}$ (equation (1)).

The following lemma moreover tells us, that each point on the medial axis is a centre of a maximal empty weakly tangent ball.

**Lemma 12** (Surjectivity on $\text{ax}(S)$) For any point $x \in \text{ax}(S)$ and $p \in \pi_S(x)$, there exists a vector $u \in \text{UBP}(p, S)$ such that $\pi_{\text{ax},S}(p, u) = x$. In other words, $B(x, |x-p|)$ is a maximally empty weakly tangent ball. Moreover, we have that

$$ \text{ax}(S) \subseteq \pi_{\text{ax},S}(\text{UBP}(S)) \subseteq \overline{\text{ax}(S)}. $$

*Proof* Let $Q = \pi_S(x)$ be the subset of $S$ that is closest to $x$. Because $x \in \text{ax}(S)$, $Q$ contains at least two points, one of them being $p$. We write $\lambda = |x - p|$. Since $S$ and $\text{ax}(S)$ are disjoint, $\lambda > 0$, and thus we can define $u = \frac{x-p}{\lambda}$.

Since the interior of the ball $B(x, \lambda)$ does not intersect $S$, it in particular does not intersect $\text{Tan}(p, S)$ and thus $B(x, \lambda)$ is weakly tangent at $p$ by Remark 7. Let us now consider the nested family $B(p + \lambda'u, \lambda')$ of weakly tangent balls at $p$. By definition, $\partial B(x, \lambda) \cap S = Q$ and therefore $B(p + \lambda'u, \lambda') \cap S = p$ for $\lambda' < \lambda$. At the same time, Lemma 8 yields that for $\lambda' > \lambda$, $p \notin \pi_S(p + \lambda'u)$. Hence the projection range in direction $u$ equals $d(p, u, \pi_S) = \lambda$ and we obtain $\pi_{\text{ax},S}(p, u) = p + \lambda u = x$ directly from Definition 11. The fact that $\pi_{\text{ax},S}(\text{UBP}(S)) \subseteq \overline{\text{ax}(S)}$ is due to Lemma 2. $\square$

We are now almost ready to state our main theorem. Before phrasing the result, we walk the reader through the assumptions and fix the notation on the way. The assumptions are illustrated in Figure 7.

**Assumption 13** We assume the following:

A1) We assume that $S$ is sphere bounded; in other words, $S = S' \cup \partial B(r) = S' \cup S(r)$, with $S' \subset B$.
A2) We consider a $C^1$ diffeomorphism $F: \mathbb{R}^d \to \mathbb{R}^d$ such that the Lipschitz constants of $F$ and $F^{-1}$ are bounded by $L_F$, and the Lipschitz constants of the differentials $DF$ and $DF^{-1}$ are bounded by $L_{DF}$. We call such a diffeomorphism a $C^{1,1}$ diffeomorphism.

A3) We further assume that the map $F$ leaves the bounding sphere $S(r)$ invariant, that is, $F(S(r)) = S(r)$.

We also use the following notation:

N1) We pick a point $c \in \text{ax}(S)$, a point $p \in \pi_S(c)$, and write $\rho = |c - p|$. Observe that since $S \cap \text{ax}(S) = \emptyset$, $\rho$ is positive. By Lemma 12, the ball $B(c, \rho)$ is a maximal empty weakly tangent ball to $S$ at $p$. Moreover, we define $u = \frac{c-p}{|c-p|}$ and note that $u \in \text{UBP}(p, S)$.

N2) We denote the tangent hyperplane to the boundary sphere of $B(c, \rho)$ at $p$ by $p + T$. The hyperplane $T$ is the orthocomplement of the vector $u: T = u^\perp$.

N3) We work with the unit vector at $F(p)$ that points inside the image of the ball $B(c, \rho)$ and is orthogonal to the hyperplane $D_p F(T)$. We denote this vector by $u'$.

The following properties of $u$ and $u'$ play an important rule in the proof of the theorem:

**Claim 14**

$$u' = \frac{(D_p F^t)^{-1}(u)}{|(D_p F^t)^{-1}(u)|}, \qquad (2)$$

where $D_p F^t$ is the transpose matrix (or the adjoint operator) of $DF$ at the point $p$, defined by

$$\forall v_1, v_2, \quad \langle v_1, D_p F(v_2) \rangle = \langle D_p F^t(v_1), v_2 \rangle.$$

**Claim 15** Let $\|D_p F - \text{Id}\| \le \varepsilon < 1$. Then the angle $\angle u, u'$ between the vectors $u$ and $u'$ satisfies

$$\cos \angle u, u' \ge \sqrt{1 - \varepsilon^2}.$$

The proofs of both of these claims are exercises in linear algebra, which we defer to the appendix.

**Theorem 16** Under the above assumptions, there exists a maximal empty weakly tangent ball $B(c', \rho')$ to the set $F(S)$, whose boundary sphere has an internal normal $u'$. In particular, the ball $B(c', \rho')$ is tangent to the affine hyperplane $F(p) + D_p F(T)$. Its radius $\rho'$ is bounded by $\rho' \in \left[ \frac{\rho}{(L_F)^3 + \rho L_{DF}(L_F)^2}, \frac{(L_F)^3 \rho}{1 - \rho L_{DF}(L_F)^2} \right]$. Assume, moreover, that the distortions of both $F$ and $DF$ are bounded, that is, for all $x \in \mathbb{R}^d$,

$$|F(x) - x| \le \varepsilon_1, \quad \|DF_x - \text{Id}\| \le \varepsilon_2 < 1, \qquad (3)$$
and $r \cdot L_{DF}(L_F)^2 \le 1/2$. Define

$$C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2) = 2r\sqrt{1 + (L_F)^6(1 + 4rL_{DF}(L_F)^2)^2 - 2(L_F)^3(1 + 4rL_{DF}(L_F)^2)\sqrt{1 - (\varepsilon_2)^2} + \varepsilon_1} \quad (4)$$

then the map $\pi_{ax,S}$ satisfies

$$|\pi_{ax,S}(p, u) - \pi_{ax,F(S)}(F(p), u')| \le C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2). \quad (5)$$

Thus, the Hausdorff distance between the medial axes of $S$ and its image $F(S)$ is bounded by

$$d_H(\operatorname{ax}(S), \operatorname{ax}(F(S))) \le C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2). \quad (6)$$

The bound $|F(x) - x| \le \varepsilon_1$ really is necessary, because we want our theorem to accommodate for rotations and translations, which rotate and translate the medial axis without changing distances and hence have Lipschitz constant 1. We further stress that if the diffeomorphism $F$ is close to the identity, its Lipschitz constant satisfies $L_F \ge 1$, because by assumption $F$ leaves the bounding sphere $S(r)$ invariant, and $L_{DF}$ is close to zero. Under further assumptions, expression (4) can be bound explicitly, as is done in Remark 18.

**Proof of Theorem 16** We first derive the bounds for the radius $\rho'$. As the first step, we apply Theorem 5 to the boundary sphere $S(c, \rho)$ of the maximal empty weakly tangent ball $B(c, \rho)$. In particular, we can choose the constant $s$ in Theorem 2.6 arbitrarily large, and the constant $t$ arbitrarily close to the reach $\text{rch}(S(c, \rho)) = \rho$, to obtain:

$$\text{rch}(F(S(c, \rho))) \ge \frac{1}{\left(\frac{L_F}{\rho} + L_{DF}\right)(L_F)^2} = \frac{\rho}{(L_F)^3 + \rho L_{DF}(L_F)^2} =: \rho_1.$$

This means that no open ball of radius $\rho_1$ tangent to the set $F(S(c, \rho))$ actually intersects $F(S(c, \rho))$. In addition, since the set $F(B(c, \rho))$ does not contain any points of $F(S)$ in its interior, no ball of radius $\rho_1$ that is tangent to $F(S(c, \rho))$ and whose centre lies inside $F(S(c, \rho))$ contains any point of $F(S)$.

The unit vector $u' \in DF_p(T)^\perp$ (defined in (2)) is defined such that the point $F(p) + \rho_1 u'$ lies inside the distorted ball $F(B(c, \rho))$. Due to the above observation, the ball $B(F(p) + \rho_1 u', \rho_1)$ is weakly tangent to $F(S)$ at $F(p)$ and contains no points of $F(S)$ in its interior.

Let us now consider the weakly tangent ball $B(F(p) + \rho''u', \rho'')$, whose radius $\rho''$ satisfies

$$\rho'' > \frac{(L_F)^3 \rho}{1 - \rho L_{DF}(L_F)^2} =: \rho_2.$$
To shorten the notation, we set

$$F(p) + \rho''u' =: c''.$$

To derive a contradiction, we assume that $B(c'', \rho'')$ is maximal empty. This is equivalent to assuming that $\text{int}B(c'', \rho'') \cap F(S) = \emptyset$, and thus $B(c'', \rho'')$ is a maximal empty weakly tangent ball to $F(p)$. Similarly to the beginning of the proof, we now apply Theorem 5 to the map $F^{-1}$ and the boundary sphere $S(c'', \rho'') = \partial B(c'', \rho'')$. As a result, the reach of $F^{-1}(S(c'', \rho''))$ is at least

$$\text{rch}\left(F^{-1}(S(c'', \rho''))\right) \geq \frac{\frac{(L_F)^3 \rho}{1 - \rho L_{DF}(L_F)^2}}{(L_F)^3 + \frac{(L_F)^3 \rho}{1 - \rho L_{DF}(L_F)^2} L_{DF}(L_F)^2} = \rho.$$

We conclude that there exists a ball that is tangent to the set $F^{-1}(S(c'', \rho''))$ at $F^{-1}(F(p)) = p$, whose radius is larger than $\rho$, and that does not contain any points of $S$ in its interior. This contradicts the fact that the ball $B(c, \rho)$ is maximal empty, and completes the proof of the first part of the statement.

We now prove the bounds on the distortion of the map $\pi_{ax,S}$.

Let $\rho' \in [\rho_1, \rho_2]$ be the radius of the maximal empty weakly tangent ball at $F(p)$ in the direction $u'$, and write $c' := F(p) + \rho'u'$ for its centre.

We stress that, as a consequence of Lemma 2 (Theorem 4.8(6) of Federer (1959)), $c' \in \text{ax}(F(S))$, but it is not necessarily true that $c' \in \text{ax}(F(S))$.

The goal is to estimate the distance between the two centres $c = \pi_{ax,S}(p, u)$ and $c' = \pi_{ax,S}(F(p), u')$. Indeed, since $c - p = \rho u$ and $c' - F(p) = \rho' u'$,

$$|c - c'| = |c - p + p - F(p) + F(p) - c'| = |\rho u + p - F(p) - \rho' c'| \\ \leq |\rho u - \rho' u'| + |F(p) - p|.$$

Due to the assumptions of the theorem, we have that $|F(p) - p| \leq \varepsilon_1$. Furthermore, thanks to Claim 15,

$$|\rho u - \rho' u'|^2 = \rho^2 + (\rho')^2 - 2\rho\rho' \cos \angle u, u' \leq \rho^2 + (\rho')^2 - 2\rho\rho' \sqrt{1 - (\varepsilon_2)^2}.$$

Recalling that $\rho' \in [\rho_1, \rho_2]$, we thus obtain

$$|\rho u - \rho' u'| \\ \leq \max \left( \sqrt{\rho^2 + (\rho_1)^2 - 2\rho \rho_1 \cos(\arcsin(\varepsilon_2))}, \sqrt{\rho^2 + (\rho_2)^2 - 2\rho \rho_2 \cos(\arcsin(\varepsilon_2))} \right) \\ = \max \left( \sqrt{\rho^2 + (\rho_1)^2 - 2\rho \rho_1 \sqrt{1 - (\varepsilon_2)^2}}, \sqrt{\rho^2 + (\rho_2)^2 - 2\rho \rho_2 \sqrt{1 - (\varepsilon_2)^2}} \right).$$
Hence,

$$|c - c'| \le \max \left( \sqrt{\rho^2 + (\rho_1)^2 - 2\rho \rho_1 \sqrt{1 - (\varepsilon_2)^2}}, \sqrt{\rho^2 + (\rho_2)^2 - 2\rho \rho_2 \sqrt{1 - (\varepsilon_2)^2}} \right) + \varepsilon_1. \quad (7)$$

As the last step, we simplify the expression (7) (at the cost of weakening the bounds). For this, we assume that $\rho L_{DF}(L_F)^2 \le 1/2$, so that

$$\rho_1 = \frac{\rho}{(L_F)^3 + \rho L_{DF}(L_F)^2} \ge \frac{\rho}{(L_F)^3} \left(1 - \rho \frac{L_{DF}}{L_F}\right), \quad (8)$$

$$\rho_2 = \frac{(L_F)^3 \rho}{1 - \rho L_{DF}(L_F)^2} \le \rho (L_F)^3 \left(1 + 2\rho L_{DF}(L_F)^2\right), \quad (9)$$

where we used that, for $x \in [0, 1/2]$, $\frac{1}{1+x} \ge 1-x$ and $\frac{1}{1-x} \le 1+2x$. We note that both $\rho_1$ and $\rho_2$ tend to $\rho$ as $L_F$ tends to 1 and $L_{DF}$ tends to 0. We now consider $|\rho_1 - \rho|$ and $|\rho_2 - \rho|$, and claim that

$$|\rho_1 - \rho|, |\rho_2 - \rho| \le \rho (L_F)^3 \left(1 + 2\rho L_{DF}(L_F)^2\right) - \rho.$$

For $|\rho_2 - \rho| = \rho_2 - \rho$, the claim holds thanks to (9). To establish this for $|\rho_1 - \rho|$ requires a small calculation:

$$|\rho_1 - \rho| = \rho - \rho_1 \le \rho - \frac{\rho}{(L_F)^3} \left(1 - \rho \frac{L_{DF}}{L_F}\right) \quad (\text{due to (8)})$$

$$\le \rho (L_F)^3 \left(1 + 2\rho L_{DF}(L_F)^2\right) - \rho \quad (\text{assuming the claim holds})$$

$$2\rho \le \rho(L_F)^3 \left(1 + 2\rho L_{DF}(L_F)^2\right) + \frac{\rho}{(L_F)^3} \left(1 - \rho \frac{L_{DF}}{L_F}\right) \quad (\text{reformulating the previous inequality})$$

$$2 \le (L_F)^3 + \frac{1}{(L_F)^3} + 2\rho L_{DF}(L_F)^5 - \rho \frac{L_{DF}}{(L_F)^4},$$

where the final inequality holds because $x^3 + x^{-3} \ge 2$, and $2x^5 - x^{-4} \ge 0$, for $x \ge 1$. We now consider the function

$$\begin{aligned} f(\delta) &= \rho^2 + \rho^2(1+\delta)^2 - 2\rho^2(1+\delta)\sqrt{1-(\varepsilon_2)^2} \\ &= \rho^2\left(\delta^2 + 2\left(1 - \sqrt{1-(\varepsilon_2)^2}\right)\delta + 2\left(1 - \sqrt{1-(\varepsilon_2)^2}\right)\right). \end{aligned}$$
The function $f$ is a second order polynomial in $\delta$ and because all coefficients are positive, the maximum of $f$ on the interval $[-\delta_m, \delta_m]$ is attained at $\delta_m$, that is,

$$ \sup_{\delta \in [-\delta_m, \delta_m]} f(\delta) = f(\delta_m). $$

By combining these results, we see that

$$ \begin{aligned} & |c - c'| \\ & \le \sqrt{f((L_F)^3 (1 + 2\rho L_{DF}(L_F)^2) - 1) + \varepsilon_1} \\ & = \sqrt{\rho^2 + (\rho(L_F)^3 (1 + 2\rho L_{DF}(L_F)^2))^2 - 2\rho (\rho(L_F)^3 (1 + 2\rho L_{DF}(L_F)^2)) \sqrt{1 - (\varepsilon_2)^2}} \\ & \qquad + \varepsilon_1 \\ & = \rho \sqrt{1 + (L_F)^6 (1 + 2\rho L_{DF}(L_F)^2)^2 - 2(L_F)^3 (1 + 2\rho L_{DF}(L_F)^2) \sqrt{1 - (\varepsilon_2)^2} + \varepsilon_1}. \end{aligned} $$

Because both $f(\delta)$ and the bound (9) are monotone in $\rho$, and $\rho$ is bounded by the radius $r$ of the bounding sphere $S(r)$, we conclude that

$$ |c - c'| \le 2r \sqrt{1 + (L_F)^6 (1 + 4r L_{DF}(L_F)^2)^2 - 2(L_F)^3 (1 + 4r L_{DF}(L_F)^2) \sqrt{1 - (\varepsilon_2)^2} + \varepsilon_1}. \quad (10) $$

For every point $c$ in $ax(S)$ we have found a point $c'$ in $\overline{ax(F(S))}$ whose distance is bounded by (10), and therefore the one-sided Hausdorff distance between the two medial axes $ax(S)$ and $\overline{ax(F(S))}$ is bounded by the same quantity. Because the symmetrical formulation of the statement, the same bound holds for the Hausdorff distance. □

## 4 Quantifying $C^{1,1}$ diffeomorphisms as deviations from identity

In this section we reformulate the main result in terms of norms on Banach spaces. This reformulation offers a more theoretical insight, and we believe the reformulated bounds are easier to work with in certain applications. Indeed, in the context of practical numerical computations, a bound on the Lipschitz constant of an operator — or, at least, a modulus of continuity — allows to control the condition number. This control is particularly useful when we calculate with objects such as the medial axis, whose (numerical) stability is often problematic in practice.

As we will see below, for this reformulation we somewhat strengthen our assumptions.

We decompose a diffeomorphism $F$ into the identity map $1_{\mathbb{R}^d}$ on $\mathbb{R}^d$, and a displacement field $\varphi: F = 1_{\mathbb{R}^d} + \varphi$. For the choice of the displacement field, we restrict
ourselves to the vector space $\mathcal{U}$ of all $C^{1,1}$ maps $\varphi$ from $\mathbb{R}^d$ to $\mathbb{R}^d$ whose restriction to the exterior $\mathbb{R}^d \setminus B(r)$ of a certain bounding ball $B(r)$ equals 0.<fnref n="3" />

A natural norm associated to $\mathcal{U}$ is one that makes it a Banach space. A typical choice, inherited from general Banach spaces of $C^{1,1}$ functions, would be for example, for $\varphi \in \mathcal{U}$,

$$ \|\varphi\|_{C^{1,1}} = \max (\|\varphi\|_{\infty}, \|D\varphi\|_{\infty}, \mathrm{Lip}(D\varphi)) . \quad (11) $$

Here we used the following notation:

**N4)** $\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}^d} |\varphi(x)|$ denotes the sup norm on $x \mapsto |\varphi(x)|$, where $|\cdot|$ is the Euclidean norm in $\mathbb{R}^d$,

**N5)** $\|D\varphi\|_{\infty} = \sup_{x \in \mathbb{R}^d} \|D\varphi(x)\|$ denotes the sup norm on $x \mapsto \|D\varphi(x)\|$, where $\|D\varphi(x)\|$ is the operator norm induced by the Euclidean norm on $\mathbb{R}^d$.

**N6)** We write $\mathrm{Lip}(D\varphi)$ for the Lipschitz semi-norm of $D\varphi$. The Lipschitz semi-norms of $\varphi$ and $D\varphi$ are defined as

$$ \mathrm{Lip}(\varphi) = \sup_{x,y \in \mathbb{R}^d, x \neq y} \frac{|\varphi(y) - \varphi(x)|}{|y - x|}, $$

and

$$ \mathrm{Lip}(D\varphi) = \sup_{x,y \in \mathbb{R}^d, x \neq y} \frac{\|D\varphi(y) - D\varphi(x)\|}{|y - x|}. $$

The norm defined in (11) makes $\mathcal{U}$ into a Banach space, since every Cauchy sequence in $\mathcal{U}$ has a limit in $\mathcal{U}$. In addition, any function $\varphi \in \mathcal{U}$ satisfies:

$$ \mathrm{Lip}(\varphi) = \|D\varphi\|_{\infty}, \quad (12) $$

$$ \|D\varphi\|_{\infty} \le r\mathrm{Lip}(D\varphi), \quad (13) $$

$$ \|\varphi\|_{\infty} \le r\mathrm{Lip}(\varphi) \le r^2\mathrm{Lip}(D\varphi), \quad (14) $$

since the restriction of $\varphi$ to $\mathbb{R}^d \setminus B(r)$ is 0. This in turn yields that $\mathrm{Lip}(D\varphi) \le \|\varphi\|_{C^{1,1}} \le \max(1, r, r^2)\mathrm{Lip}(D\varphi)$. Thus, in $\mathcal{U}$, the norm $\varphi \mapsto \mathrm{Lip}(D\varphi)$ is equivalent to the norm $\varphi \mapsto \|\varphi\|_{C^{1,1}}$.

We can now state slightly less general version of Theorem 16 in terms of the Banach space $(\mathcal{U}, \varphi \mapsto \mathrm{Lip}(D\varphi))$.

**Theorem 17** Let $S$ satisfy **A1**). That is, $S \subseteq \mathbb{R}^d$ be sphere bounded, or in other words, $S = S' \cup S(r)$, where $S'$ is a subset of the ball $B(r)$ of radius $r > 0$. The assumption **A3**) is strengthened to:

**A3\***) Let further $F$ be a $C^{1,1}$ diffeomorphism from $\mathbb{R}^d$ to itself that leaves the set $\mathbb{R}^d \setminus B(r)$ invariant.

<footnote n="3">³ This is more restrictive than assuming that the restriction to the bounding sphere $S(r)$ is 0, but it simplifies matters in this section.</footnote>
Define

two displacement fields $\varphi, \tilde{\varphi} \in \mathcal{U}$ such that $F = 1_{\mathbb{R}^d} + \varphi$ and

$$(1_{\mathbb{R}^d} + \tilde{\varphi}) \circ (1_{\mathbb{R}^d} + \varphi) = 1_{\mathbb{R}^d}.$$

Finally, let $\varepsilon = \max(\mathrm{Lip}(D\varphi), \mathrm{Lip}(D\tilde{\varphi}))$.

If $r\varepsilon \le 1/4$, the Hausdorff distance between the medial axes of the set $S$ and its image $F(S)$ is bounded by $d_H(\mathrm{ax}(S), \mathrm{ax}(F(S))) \le (1 + \sqrt{50}) r^2\varepsilon + \mathcal{O}(r^3\varepsilon^2)$. In particular, $d_H(\mathrm{ax}(S), \mathrm{ax}(F(S))) = \mathcal{O}(r^2\varepsilon)$.

**Proof** We denote $L_\varphi = \mathrm{Lip}(\phi)$. Expressions (12), (13) and (14) yield:

$$L_{\varphi} \le r\varepsilon, \quad L_{DF} = \varepsilon, \quad L_F \le 1 + L_{\varphi} \le 1 + r\varepsilon, \quad \varepsilon_1 \le r^2\varepsilon, \quad \varepsilon_2 \le r\varepsilon. \tag{15}$$

We deduce

$$r\varepsilon \le 1/4 \implies r\varepsilon(1+r\varepsilon)^2 \le 1/2 \implies rL_{DF}(L_F)^2 \le 1/2.$$

Thus, the conditions of Theorem 16 are satisfied. Next, we reformulate the inequality (6) of Theorem 16. The expression $E$ under the square root at the right hand side of this inequality is:

$$E = 1 + (L_F)^6 \left(1 + 4rL_{DF}(L_F)^2\right)^2 - 2(L_F)^3 \left(1 + 4rL_{DF}(L_F)^2\right) \sqrt{1 - (\varepsilon_2)^2}. \tag{16}$$

By replacing $L_F$ by $1 + L_\varphi$ in $E$, the constants, as well as the degree-one terms in $L_\varphi$, $rL_{DF}$, and $\varepsilon_2$, cancel out. More precisely,

$$E = 16r^2 L_{DF}^2 + \varepsilon_2^2 + 24rL_\varphi L_{DF} + 9L_\varphi^2 + \mathcal{O}(|rL_{DF}, L_\varphi, \varepsilon_2|^3). \tag{17}$$

Finally, by substituting inequalities (15) into (17), we obtain

$$E \le 50r^2\varepsilon^2 + \mathcal{O}(r^3\varepsilon^3),$$

and

$$d_H(\mathrm{ax}(S), \mathrm{ax}(F(S))) \le (1 + \sqrt{50}) r^2\varepsilon + \mathcal{O}(r^3\varepsilon^2).$$

□

**Remark 18** We can also bound the expression $E$ from equation (16) explicitly. Indeed, if $0 \le x \le 1/4$ then $-\sqrt{1-x^2} \le -1+4(4-\sqrt{15})x^2$, and

$$E \le 50.02(r\varepsilon)^2 + 329.11(r\varepsilon)^3 + 1126.37(r\varepsilon)^4 + 2487.66(r\varepsilon)^5 + 3841.65(r\varepsilon)^6$$

<figure><img src="image_10.png" /></figure>
$$+ 4276.32(r_{\varepsilon})^7 + 3428.06(r_{\varepsilon})^8 + 1928(r_{\varepsilon})^9 + 720(r_{\varepsilon})^{10} + 160(r_{\varepsilon})^{11} + 16(r_{\varepsilon})^{12}, \quad (18)$$

where we still assume that $r_{\varepsilon} \le 1/4$.

**Remark 19** Observe that the bound $O(r^2\varepsilon)$ is consistent with a scaling by factor $\lambda$: $S \mapsto \lambda S$, $F(\cdot) \mapsto \lambda F(\cdot/\lambda)$. Under such a scaling, the radius $r$ is multiplied by $\lambda$, while the Lipschitz constant $\mathrm{Lip}(D\varphi)$ — and therefore $\varepsilon$ — is divided by $\lambda$. Furthermore, the Hausdorff distance $d_H(\mathrm{ax}(S), \mathrm{ax}(F(S)))$ increases by a factor $\lambda$. By considering a diffeomorphism that translates the set $S \setminus S(r)$ while keeping the bounding sphere $S(r)$ fixed, we see that this bound is asymptotically optimal.

# 5 Alternative assumptions

In this section we replace the assumptions **A1)** and **A3)** from Assumption **13** with alternative assumptions that imply the stability of the medial axis in one form or other.

We recall that assumptions **A1)** and **A3)** from Assumption **13** suppose that the set $S$ is sphere bounded, and that the ambient diffeomorphism $F$ preserves the bounding the bounding sphere of radius $r$.

## 5.1 Global density assumptions

First, we assume bounds on the density of $S$ as well as $F(S)$ in $\mathbb{R}^d$. To be more concrete, instead of assumptions **A1)** and **A3)** from Assumption **13**, we assume that:

**A1')** $S$ is dense in $\mathbb{R}^d$, that is,

$$d_H(S, \mathbb{R}^d) \le r. \qquad (19)$$

**A3')** The density is preserved under the ambient diffeomorphism, that is,

$$d_H(F(S), \mathbb{R}^d) \le r. \qquad (20)$$

We can now state the following corollary, which is not so much a corollary of Theorem **16**, but rather of its proof.

**Corollary 20** The statement of Theorem **16** still holds under the assumptions **A1')** and **A3')** instead of **A1)** and **A3)**.

*Proof* The variable $r$ plays no material role in the proof of Theorem **16** until the final two paragraphs of the proof. From there, it is used as an upper bound for the radius $\rho$ of the maximal empty weakly tangent ball. The assumptions **A1')** and **A3')** provide exactly the same bound $r$ on $\rho$. The result then follows in an identical manner. $\square$

## 5.2 Local density assumptions

In this subsection, we make no assumption on the ambient diffeomorphism, and instead concentrate on stability in a neighbourhood. This comes at the price of having to
consider the two one-sided Hausdorff distances. The reason for this is simple: even when two sets are close in terms of the Hausdorff distance, the Hausdorff distance between the intersections of both sets with the same ball is not necessarily small. In fact, the distance may be infinite, since the one intersection can be empty while the other might not be.

Instead of assumptions **A1** and **A3**) from Assumption 13, we assume that:

**A1")** The set $S$ is dense near the origin, that is,

$$d_H(S \cap B(2r), B(2r)) \leq r.$$

**A3")** The density of $S$ is preserved under the diffeomorphism, that is,

$$d_H(F(S) \cap B(2r), B(2r)) \leq r.$$

Also in this context we have a corollary of Theorem 16, or rather of its proof.

**Corollary 21** *The statement of Theorem 16 still holds after replacing assumptions **A1**) and **A3**) by assumptions **A1")** and **A3")**, with the following modifications:*

* The bound (5) holds only for points $p \in B(r)$.

* The bound (6) is replaced by the following two bounds on the one-sided Hausdorff distance:

$$ \begin{aligned} d_H^o(\operatorname{ax}(S) \cap B(r), \operatorname{ax}(F(S)) \cap B(2r)) &\leq C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2) \\ d_H^o(\operatorname{ax}(F(S)) \cap B(r), \operatorname{ax}(S) \cap B(2r)) &\leq C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2) \end{aligned} \quad (21) $$

or equivalently,

$$ \begin{aligned} d_H^o(\operatorname{ax}(S) \cap B(r), \operatorname{ax}(F(S))) &\leq C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2) \\ d_H^o(\operatorname{ax}(F(S)) \cap B(r), \operatorname{ax}(S)) &\leq C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2). \end{aligned} \quad (22) $$

* We assume that $C_L(r, L_F, L_{DF}, \varepsilon_1, \varepsilon_2) < r/2$.

**Proof** As we pointed out in the proof of Corollary 20, the variable $r$ plays no material role in the proof of Theorem 16 until the final two paragraphs of the proof, where it is used as an upper bound for the radius $\rho$ of the maximal empty weakly tangent ball. The assumptions **A1")** and **A3")** provide the same bound on $\rho$ for all points $p \in B(r)$, and bound (5) follows. The bound (21) is an immediate consequence of (5). □

## 5.3 Lipschitz stability without boundary conditions on the diffeomorphism

In Theorem 17 we reformulated Theorem 16 in terms of a Banach space. In this subsection we aim for the same type of formulation, but we drop the assumptions **A3**) or **A3*), and instead make assumptions on the norm introduced in inequality (11), which we will recall shortly.
Let us make this statement more precise. We do not assume that the diffeomorphism $F$ leaves either the bounding sphere $S(r)$ or its exterior invariant. As before, we write $F$ (and its inverse) in terms of a displacement field $\varphi$ (and $\tilde{\varphi}$). In other words, $F = \mathbb{1}_{\mathbb{R}^d} + \varphi$ and

$$ (\mathbb{1}_{\mathbb{R}^d} + \tilde{\varphi}) \circ (\mathbb{1}_{\mathbb{R}^d} + \varphi) = \mathbb{1}_{\mathbb{R}^d}. $$

Assumption A3*) then implies that $\varphi$ and $\tilde{\varphi}$ are zero outside $S(r)$. Without these assumptions, inequalities (12), (13), and (14) do not necessarily hold.

We emphasize that, in this setting, $\varphi \mapsto \text{Lip}(D\varphi)$ is only semi-norm. However, a natural norm such as the one we defined in Section 4,

$$ \|\varphi\|_{C^{1,1}} = \max (\|\varphi\|_{\infty}, \|D\varphi\|_{\infty}, \text{Lip}(D\varphi)), \quad (11) $$

still allows us to formulate a Lipschitz stability result. We stress that if we rescale the entire setting, the first entry of the maximum in equation (11) rescales linearly, the second entry remains the same, and the third entry rescales inverse linearly. At the same time, the Hausdorff distance rescales linearly. We also note that since $\varphi$ is $C^1$, it follows trivially that $\|D\varphi\|_{\infty} = L_{\varphi}$.

Assuming $r = 1$ and observing that $\varepsilon_2 \le L_{\varphi}$, Corollary 21, and equations (4) and (17) yield:

$$ d_H^o(\text{ax}(S) \cap B(0, 1), \text{ax}(F(S))) \le (1 + \sqrt{50}) \tilde{\varepsilon} + \mathcal{O}(\tilde{\varepsilon}^2), $$

$$ d_H^o(\text{ax}(F(S)) \cap B(0, 1), \text{ax}(S)) \le (1 + \sqrt{50}) \tilde{\varepsilon} + \mathcal{O}(\tilde{\varepsilon}^2), $$

where, using the norm $\| \cdot \|_{C^{1,1}}$ defined in equation (11),

$$ \tilde{\varepsilon} := \max(\|\varphi\|_{C^{1,1}}, \|\tilde{\varphi}\|_{C^{1,1}}). $$

We can reformulate this in a way that is compatible with scaling, but at the cost of introducing a different norm:

Given some number $r > 0$, we define

$$ \|\varphi\|_{C^{1,1,r}} := \max (\|\varphi\|_{\infty}, r \|D\varphi\|_{\infty}, r^2 \text{Lip}(D\varphi)). $$

This norm is homogeneous in the sense that

$$ \|x \mapsto r \varphi (\frac{1}{r} x)\|_{C^{1,1},r} = r \|\varphi\|_{C^{1,1},1} = r \|\varphi\|_{C^{1,1}}. $$

Thus, we have the following:

$$ d_H^o(\text{ax}(S) \cap B(0, r), \text{ax}(F(S))) \le (1 + \sqrt{50}) \tilde{\varepsilon}_r + \mathcal{O}(\tilde{\varepsilon}_r^2), $$

$$ d_H^o(\text{ax}(F(S)) \cap B(0, r), \text{ax}(S)) \le (1 + \sqrt{50}) \tilde{\varepsilon}_r + \mathcal{O}(\tilde{\varepsilon}_r^2), $$
where

$$ \tilde{\epsilon}_r := \max(\|\varphi\|_{C^{1,1,r}}, \|\tilde{\varphi}\|_{C^{1,1,r}}). $$

We note that only the behaviour of $\varphi$ and $\tilde{\varphi}$ inside $B(0, 2r)$ is relevant in this setting, so that the suprema over the whole of $\mathbb{R}^d$ defined in **N4**) and **N5**) can be replaced by suprema over $B(0, 2r)$. By extension, the $\| \cdot \|_{\infty}$ norms can be weakened in the same manner.

## 6 Conclusion and future work

We proved the Hausdorff stability of the medial axis of a closed set, without any further assumption on the nature of the set except that it is either bounded or (locally) sufficiently dense.

With regard to applications, our result is the first step towards providing a provably correct image recognition in the context of numerous scientific disciplines, and in particular of astrophysics. The next step is to produce physics-informed models for the medial axis as occurring in astronomical data.

On the mathematical side, we conclude with a conjecture generalizing our result to compact Riemannian manifolds with bounded curvature.

**Conjecture 22** Let $\mathcal{N}$ be a compact Riemannian manifold with bounded sectional curvature$^4$ and $S$ a closed subset of $\mathcal{N}$. Then the generalized medial axis of $S$ in $\mathcal{N}$, called the cut locus (Attali et al. 2024; Kapovitch and Lytchak 2021), is Lipschitz stable under diffeomorphisms of $\mathcal{N}$.

## A Proofs of the Claims

**Proof of Claim 14** Since any invertible matrix $A$ satisfies $(A^t)^{-1} = (A^{-1})^t$, one has:

$$ \begin{align*} w \in D_p F(u^{\perp}) &\iff \langle D_p F^{-1}(w), u \rangle = 0 \\ &\iff \langle w, (D_p F^{-1})^t u \rangle = 0 \\ &\iff \langle w, u' \rangle = 0 \\ &\iff w \in u'^{\perp}, \end{align*} $$

and thus

$$ D_p F(u^{\perp}) = u'^{\perp}. \qquad (23) $$

In other words, we have shown that $u'$ is orthogonal to $D_p F(u^{\perp}) = D_p F(T)$.

<footnote>$^4$ See Berger (2003) for definitions and a pedagogical overview of the properties of these manifolds.</footnote>
<figure><img src="image_11.png" /><figcaption>**Fig. 8** Since $\|v - D_p F(v)\| \le \varepsilon |v|$, the vector $D_p F(v)$ lies in the green ball $B(v, \varepsilon |v|)$. Since $\varepsilon < 1$, the angle between $v$ and $D_p F(v)$ is upper-bounded by $\arcsin \varepsilon < \pi/2$</figcaption></figure>

Because

$$ \langle D_p F(u), (D_p F^{-1})^t(u) \rangle = \langle D_p F^{-1}(D_p F(u)), u \rangle = \langle u, u \rangle > 0, $$

we deduce that $\langle D_p F(u), u' \rangle > 0$. This is in turn equivalent to $u'$ pointing towards the interior of $F(B(c, \rho))$. $\square$

**Proof of Claim 15** We first show that $\angle u, u' < \pi/2$. Indeed, define the vector $w$ as

$$ w = (D_p F^t)^{-1}(u), $$

that is, the vector satisfying $u = D_p F^t(w)$. Then $u' = \frac{w}{|w|}$ (see equation (2)), and

$$ \begin{aligned} |w| \langle u, u' \rangle &= \langle u, w \rangle = \langle D_p F^t w, w \rangle && (\text{because, by assumption, } \|DF_p - \text{Id}\| < 1) \\ &= \langle w, D_p F w \rangle = |w|^2 + \langle w, (D_p F - \text{Id})w \rangle \\ &\ge |w|^2 - |w|^2 \|D_p F - \text{Id}\| \\ &> 0. \end{aligned} $$

Thus, $\langle u, u' \rangle > 0$, and therefore $\angle u, u' < \pi/2$.

Furthermore, consider a vector $v \in u^{\perp}$. Since $\|v - D_p F(v)\| \le \|D_p F - \text{Id}\| |v| \le \varepsilon|v|$, the angle between $v$ and $D_p F(v)$ is upper-bounded by $\arcsin \varepsilon < \pi/2$, as illustrated in Figure 8. This yields a bound on the angle between the tangent spaces $u^{\perp}$ and $D_p F(u^{\perp})$:

$$ \sin \angle u^{\perp}, D_p F(u^{\perp}) = \sin \sup_{v \in u^{\perp}, w \in D_p F(u^{\perp})} \angle v, w \le \varepsilon. \quad (24) $$

Using (23) and (24) we deduce that:

$$ \sin \angle u, u' = \sin \angle u^{\perp}, u'^{\perp} \le \varepsilon. $$

Finally, since $\angle u, u' < \pi/2$, $\cos \angle u, u' \ge \sqrt{1 - \varepsilon^2}$. This concludes the proof. $\square$
## B The Calculation for inequality (18)

As mentioned, using that $-\sqrt{1-x^2} \le -1 + 4(4 - \sqrt{15})x^2$ if $0 \le x \le 1/4$, we find that $E$ is upper bounded by

$$ 
\begin{aligned}
E &\le 1 + (1 + r\epsilon)^6 (1 + 4r\epsilon(1 + r\epsilon)^2)^2 - 2(1 + r\epsilon)^3 (1 + 4r\epsilon(1 + r\epsilon)^2) \\
&\quad + 2(1 + r\epsilon)^3 (1 + 4r\epsilon(1 + r\epsilon)^2) 4(4 - \sqrt{15})(r\epsilon)^2 \\
&= 49(r\epsilon)^2 + 322(r\epsilon)^3 + 1103(r\epsilon)^4 + 2446(r\epsilon)^5 + 3801(r\epsilon)^6 + 4256(r\epsilon)^7 \\
&\quad + 3424(r\epsilon)^8 + 1928(r\epsilon)^9 + 720(r\epsilon)^{10} + 160(r\epsilon)^{11} + 16(r\epsilon)^{12} \\
&\quad + (4 - \sqrt{15}) (8(r\epsilon)^2 + 56(r\epsilon)^3 + 184(r\epsilon)^4 + 328(r\epsilon)^5 + 320(r\epsilon)^6 + 160(r\epsilon)^7 \\
&\quad + 32(r\epsilon)^8) \\
&= 50.02(r\epsilon)^2 + 329.11(r\epsilon)^3 + 1126.37(r\epsilon)^4 + 2487.66(r\epsilon)^5 + 3841.65(r\epsilon)^6 \\
&\quad + 4276.32(r\epsilon)^7 + 3428.06(r\epsilon)^8 + 1928(r\epsilon)^9 + 720(r\epsilon)^{10} + 160(r\epsilon)^{11} + 16(r\epsilon)^{12},
\end{aligned}
\quad (18) $$

where we still assume that $r\epsilon \le 1/4$.

## C Federer's Tubular Neighbourhood Lemma

We recall:

**Lemma 23** (Federer's tubular neighbourhood lemma, Theorem 4.8 (12) of Federer (1959)) Let $p \in S$ and $lfs(p) > 0$. The generalized normal space to $S$ at $p$ is characterized by the following property: For any $\rho \in \mathbb{R}$ satisfying $0 < \rho < lfs(p)$,

$$ \mathrm{Nor}(p, S) = \{\lambda v \in \mathbb{R}^d \mid \lambda \ge 0, |v| = \rho, \pi_S(p + v) = \{p\}\}. $$

In particular, $\mathrm{Nor}(p, S)$ is a convex cone. The generalized tangent space $\mathrm{Tan}(p, S)$ is the convex cone dual to $\mathrm{Nor}(p, S)$.

**Acknowledgements** We are greatly indebted to Fred Chazal for sharing his insights. We thank the reviewers of both this journal and the Symposium on Computational Geometry (where an extended abstract was presented), their comments improved the presentation of the paper. We further thank Erin Chambers, Christopher Fillmore, and Elizabeth Stephenson for early discussions and all members of the Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape team (Inria) for the atmosphere in which this research was conducted.

**Funding** Open Access funding enabled and organized by Projekt DEAL. The third author was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411 and the Austrian science fund (FWF) grant No. M-3073, the ANR grant StratMesh (ANR-24-CE48-1899-02), and the welcome package from IDEX of the Université Côte d'Azur, ANR-15-IDEX-01.

This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, DFG project No. 524578210 and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.
**Availability of data and materials** Not applicable, i.e. no (synthetic) data was collected (generated).

**Declarations**

**Conflicts of Interest/Competing Interests** None.

**Ethical Approval** Not applicable.

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