# Geodesics of length less than $\pi R$ in a set of reach R are unique and continuous with respect to the end points

André Lieutier, Mathijs Wintraecken

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André Lieutier, Mathijs Wintraecken. Geodesics of length less than $\pi R$ in a set of reach R are unique and continuous with respect to the end points. 2025. hal-05368564

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# Geodesics of length less than $\pi R$ in a set of reach $R$ are unique and continuous with respect to the end points

André Lieutier ⑸

No affiliation

[Aix-en-Provence, France]

Mathijs Wintraecken ⑸ ID

Inria centre Université Côte d'Azur

[Sophia Antipolis, France]

## Abstract

Positive reach underpins many results in computational geometry and topology. It is used for triangulation criteria, topological inference, and manifold learning. The geometric properties of these sets have therefore been studied intensely. Here we focus on the shortest paths or geodesics in these sets. Our main result states that geodesics of length strictly less than $\pi R$ in a set of reach $R$ are unique. This in turn implies that such geodesics are continuous with respect to the end points.

**2012 ACM Subject Classification** Theory of computation → Computational geometry

**Keywords and phrases** Reach, geodesics, metric geometry

**Related Version** A full version of this paper is available at https://hal.inria.fr/hal-????????

**Funding** Mathijs Wintraecken: Supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411. The Austrian science fund (FWF) M-3073. The welcome package from IDEX of the Université Côte d'Azur, ANR-15-IDEX-01. The French National Research Agency (ANR) under grant StratMesh, ANR-24-CE48-1899.

**Acknowledgements** We thank Jean-Daniel Boissonnat for discussion and Eddie Aamari for his insight in possible applications.

## 1 Introduction

Federer [13] introduced the reach of a closed set $S \subset \mathbb{R}^d$ as the minimum of the distance from $S$ to the medial axis $ax(S)$, i.e. the set of points in $\mathbb{R}^d$ for which the closest point in $S$ is not unique. Assumptions on the reach (and its local version the local feature size [3]) underpin the correctness of many algorithms in computational geometry and topology. In this paper we consider geodesics in this space and show that they are both unique and depend continuously on the end points if the geodesics are sufficiently short. Moreover, the bound on the length of the geodesics that implies uniqueness (and therefore continuity) is in fact tight, because it is attained on the sphere.

### 1.1 Motivation

**Triangulation algorithms** The assumption of positive reach is central to almost all triangulation algorithms, see for example [9, 11, 5], as well as manifold learning, see for example [15, 14, 2, 29]. However these algorithms often consider manifolds of positive reach instead of general sets of positive reach. If we want to extend our algorithm beyond the realm of manifolds to more general stratified spaces, starting with 'well behaved' sets of positive

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

© André Lieutier, and Mathijs Wintraecken;
licensed under Creative Commons License CC-BY 4.0

42st International Symposium on Computational Geometry (SoCG 2026).

Editors: ???? and ????; Article No.: pp.:1–11

<figure><img src="image_3.png" /></figure>

Leibniz International Proceedings in Informatics
LIPICS Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

<figure><img src="image_4.png" /></figure>
## XX:2 Geodesics of length less than $\pi R$ in a set of reach $R$ are unique and continuous

reach<fnref n="1" /> or at least sets of which the strata have positive (local [23]) reach understanding the geometry of those sets seems essential.

**Metric learning** Another motivation comes from metric learning. In [1] this is explained as follows:

> "In the data analysis area, metric learning refers to the problem of finding a distance $d$ over the space of observations that is relevant for a given task at stake [34, 31]. For instance, in a supervised framework where one is provided with tuples of allegedly similar or dissimilar observations, the goal is to find a distance that is small on the similar tuples and large on the dissimilar ones. There is a wide range of existing methods in the literature, ranging from parametric (LSI [33], MCML [17], LDML [20] among others) to nonparametric (DMLMJ [28], kernel methods [22, 8], to cite a few).

> In an unsupervised setting, metric learning aims at finding a metric that takes into account the underlying geometry of the data. That is, it amounts to estimating the shortest-path (or geodesic) distance. Often, this is done via a dimension reduction technique: any low-dimensional embedding of the data gives rise to a new distance over the data in the embedded space. Existing algorithms include PCA, t-SNE [21], MDS [10], Isomap [32], or MVU [4]. See [31] for a thorough overview of the field."

> And indeed in [1] geodesics play a crucial role in establishing optimal bounds.

## 1.2 Contribution

This paper focusses on the following result:

**Theorem 1.** Let $S$ be a set of positive reach $R$, then for any two points $a, b \in S$ such that $d_S(a, b) < \pi R$ there unique minimizing geodesic in $S$ connecting a and b.

However, the following corollary is perhaps even more important for applications:

**Corollary 2.** Let $S$ be a set of positive reach $R$, then the geodesic connecting any two points $a, b \in S$ is continuous (in the $C^0$ sense) with respect to the end points provided that $d_S(a, b) < \pi R$.

Both results are tight, because the bound is attained on the sphere.

## 1.3 Previous work

**CAT($k$) spaces** The terminology $\text{CAT}(k)$ was introduced by Gromov [19] for metric spaces that satisfy some weak form of curvature bound. We'll now give a rough definition of $\text{CAT}(k)$ spaces, but refer to [7] for a precise introduction, see Section II.1. Let $(\mathcal{M}, d_{\mathcal{M}})$ be a geodesic metric space, that is a space where the length between two points is defined as the minimum of all lengths of continuous (rectifiable) curves connecting the two points, i.e. the minimum is attained. Such a minimizing curve is called a geodesic. A geodesic triangle in such a space consists of three points and the geodesics connecting those points. We can then consider a geodesic triangle with the same edge lengths in a (simply connected) space of constant curvature $k$ (that is a sphere, Euclidean space, or hyperbolic space). This triangle is called a

<footnote n="1">¹ Sets of positive reach are stratified spaces [13], where the stratification is understood in a weak sense. However, it is not clear that sets of positive reach are triangulable in general or additional assumptions are needed. This having been said, there are compact sets of (arbitrarily large) positive reach that are not (piecewise smoothly/linearly) homeomorphic to a finite simplicial complex.</footnote>
comparison triangle. We say that diameter of a simply connected space of constant curvature is $\pi/\sqrt{k}$ if $k > 0$ and $\infty$ otherwise. We also refer to this as the diameter of the comparison space. These geodesic triangle in $M$ satisfies the CAT($k$) inequality if the distances between the points on the triangle are less than the corresponding distances between points on the comparison triangle in a space of constant curvature $k$. If this holds for any geodesics triangle of diameter less than the diameter of the comparison space, then we say that the entire space is CAT($k$).

**Sets of positive reach as CAT($k$) spaces** Lytchak [24, 25] proved sets of positive reach are CAT($k$), for some $k$. Because in (simply connected) spaces of constant curvature, geodesic triangles with small diameter and one edge length equal to zero consist of two coinciding edges, the uniqueness of sufficiently small geodesics follows directly from Lytchak's result. However, from a computational perspective the result of Lytchak is ineffective, because we only know that such a constant exists and not what this constant may be.

Another way that this paper distinguishes itself compared to the literature, is that the results are all completely elementary, in the sense that we only need some results by Federer [13] and Euler-Lagrange theory [30].

## 1.4 Future work

Many questions on the geometry of sets of positive reach and the behaviour of geodesics in sets of positive reach of course remain open. Determining the $k$ for which a set of reach $R$ is CAT($k$), would be an important next step and would provide tools to tackle subsequent problems. We conjecture that $k = 1/R$.

## 2 Result

### 2.1 Notation

We'll denote sets of positive reach by $S$ and the closest point projection by $\pi_S$. We use the notation $B(c, r)$ for a ball in Euclidean spaces with centre $c$ and radius $r$. Curves are generally denoted by $\gamma$ and most importantly:

We'll use Newton's notation for the derivatives of curves that is, $\dot{\gamma} = \frac{d}{dt}\gamma$.

### 2.2 Uniqueness

We can now prove the main result.

We first recall a convexity result from [6]:

► **Corollary 3** (Corollary 1 of [6]). Let $S \subset \mathbb{R}^d$ be a closed set with positive reach $R > 0$. Then, for any $r' < R$ and any $x \in \mathbb{R}^d$, if $B(x, r')$ is the closed ball centered at $x$ with radius $r'$, then $S \cap B(x, r')$ is geodesically convex in $S$.

► **Lemma 4.** Let $S$ have reach $R$ and assume $p, q \in S$ with $p, q \le 2R$. Define $x = \lambda p + (1-\lambda)q$. We have that

$$ 
\begin{align*}
d(x, S) &= |x - \pi_S(x)| \\
&\le R - \sqrt{R^2 - \lambda(1-\lambda)|p-q|^2} \\
&\le \frac{\lambda(1-\lambda)|p-q|^2}{2R} + \mathcal{O}\left(\left(\frac{\lambda(1-\lambda)|p-q|^2}{2R}\right)^2\right)
\end{align*} 
$$
XX:4 Geodesics of length less than $\pi R$ in a set of reach $R$ are unique and continuous

<figure><img src="image_5.png" /><figcaption>Figure 1 The region in which the geodesic connecting $p$ and $q$ lies is indicated in green, the grey dashed circles have radius $R$.</figcaption></figure>

**Proof.** Corollary 3 implies that the geodesic connecting $p$ and $q$ is contained in the intersection of all balls of radius less than the reach that contain $p$ and $q$ so that in particular the geodesic is contained in spindle, as depicted in Figure 1. This implies in turn that $|x - \pi_S(x)|$ less or equal to the distance from $x$ to the spindle. Let's restrict ourselves to the plane given by $p$, $q$, and $\pi_S(x)$ and consider a circle of radius $R$ with centre $c$ that yields the boundary of the spindle in this plane, see again Figure 1. By the cosine rule have that

$$|c - \lambda p + (1 - \lambda)q|^2 = R^2 + \lambda^2 |p - q|^2 - 2R\lambda |p - q| \cos \theta$$

where $\cos \theta = \frac{|p-q|}{2R}$ (the value for $\theta$ follows by inspection of the triangle $pc_1(p+q)$). So that

$$ \begin{aligned} |c - \lambda p + (1 - \lambda)q|^2 &= R^2 + \lambda^2 |p - q|^2 - \lambda |p - q|^2 \\ &= R^2 - (1 - \lambda)\lambda |p - q|^2. \end{aligned} $$

The first inequality now follows by observing that the distance from $x$ to the spindle is given by $R - |c - \lambda p + (1 - \lambda)q|$, see Figure 1. The second inequality follows by taking a Taylor series. ◀

We'll need the following corollary of [13, Theorem 4.8(8)]. Theorem 4.8(8) of [13] says the closest point projection $\pi_S$ at a point $x$ is $\frac{1}{1-d(x,S)/R}$-Lipschitz. We therefore immediately have:
**Corollary 5.** If $\gamma: [0, \ell] \to \mathbb{R}^d$ is a $C^1$ curve, then the length of $\pi_S(\gamma)$ is upper bounded as follows

$$ \text{length}(\pi_S(\gamma)) \le \int_0^\ell \frac{1}{1 - d(\gamma(t), S)/R} |\dot{\gamma}(t)| dt. \quad (1) $$

The following simple but very useful bound will be instrumental in the proof below.

**Lemma 6.** If a and b are two unit vectors we have that

$$ |\lambda a + (1 - \lambda)b|^2 = 1 - (1 - \lambda)\lambda|a - b|^2. $$

For vectors of arbitrary length we have

$$ |\lambda a + (1 - \lambda)b|^2 = \lambda|a|^2 + (1 - \lambda)|b|^2 - \lambda(1 - \lambda)|a - b|^2. $$

**Proof.** We have that for any two unit vectors a, b that:

$$ \begin{align} |\lambda a + (1 - \lambda)b|^2 &= \lambda^2|a|^2 + (1 - \lambda)^2|b|^2 + 2\lambda(1 - \lambda)a \cdot b \tag{2} \\ &= \lambda^2 + (1 - \lambda)^2 + 2\lambda(1 - \lambda)a \cdot b \tag{using that $|a| = |b| = 1$} \\ &= 1 - (1 - \lambda)\lambda(2 - 2a \cdot b) \end{align} $$

$$ \begin{align} |a - b|^2 &= |a|^2 + |b|^2 - 2a \cdot b \tag{3} \\ &= 2 - 2a \cdot b \tag{using that $|a| = |b| = 1$} \end{align} $$

and thus

$$ |\lambda a + (1 - \lambda)b|^2 = 1 - (1 - \lambda)\lambda|a - b|^2. $$

For a, b of arbitrary length we combine (2) and (3) as follows

$$ \begin{align} |\lambda a + (1 - \lambda)b|^2 &= \lambda^2|a|^2 + (1 - \lambda)^2|b|^2 + \lambda(1 - \lambda)(|a|^2 + |b|^2 - |a - b|^2) \\ &= \lambda|a|^2 + (1 - \lambda)|b|^2 - \lambda(1 - \lambda)|a - b|^2. \end{align} $$

**Theorem 1.** Let $S$ be a set of positive reach $R$, then for any two points $a, b \in S$ such that $d_S(a, b) < \pi R$ there unique minimizing geodesic in $S$ connecting a and b.

**Overview proof** The core idea of the proof is the following: Given two geodesics $\gamma_1$ and $\gamma_2$ connecting $a, b$ that are parametrized by arc length we have that $\lambda\gamma_1 + (1 - \lambda)\gamma_2$ is shorter than $\gamma_1$ and $\gamma_2$ because $|\lambda\dot{\gamma}_1 + (1 - \lambda)\dot{\gamma}_2| < 1$, if $\lambda \in (0, 1)$. So$^2$ if $\lambda\gamma_1 + (1 - \lambda)\gamma_2 \subset S$ we would find a contradiction, because there would be a curve connecting $a$ and $b$ that would be shorter than a geodesic. The main difficulty of the proof is that $\lambda\gamma_1 + (1 - \lambda)\gamma_2$ is generally not a $S$, but we need to project one $S$ to find a curve that is. The technical difficulty lies with dealing with this projection.

The proof, perhaps surprisingly, also relies on a result from mathematical physics. Consider the Lagrangian of the harmonic oscillator:

$$ \mathcal{L}_{\text{HO}} = |\dot{x}|^2 - \frac{|x|^2}{R^2} $$

<footnote n="2">2 Here abuse notation a little bit because we do not distinguish the notation for a geodesic and its parametrization.</footnote>
XX:6 Geodesics of length less than $\pi R$ in a set of reach $R$ are unique and continuous

Write $S_{HO}$ for the action, that is

$$ S_{HO} = \int_{0}^{\tau} \mathcal{L}_{HO} dt. \qquad (4) $$

The Euler-Lagrange equation,

$$ \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{x}} = \frac{\partial \mathcal{L}}{\partial x}, $$

in this case yields

$$ \ddot{x}(t) = -\frac{x(t)}{R^2}, $$

with the familiar solutions $c_1 \sin(t/R) + c_2 \cos(t/R) + c_3 = c_4 \sin(\frac{t-t_0}{R}) + c_3$. We have the following, see [18, Section V], [27] for this specific result<fnref n="3" /> and [16] for general theory:

► **Theorem 7** (Adjusted from [18] and [27]). *The solution to the Euler-Lagrange equation is a global minimum of the action (4) provided that $\tau < \pi R$.*

This means in particular that:

► **Corollary 8.** Let $c_4$ be a constant. If $\tau < \pi R$ and $x(t)$ is a function that satisfies $x(0) = 0$ and $x(\tau) = c_4 \sin(\frac{\tau}{R})$, then we have that

$$ \begin{aligned} S_{HO}(x) &= \int_{0}^{\tau} |\dot{x}|^2 - \frac{|x|^2}{R^2} dt \\ &\geq \int_{0}^{\tau} \left( \frac{c_4^2 \cos^2\left(\frac{t}{R}\right) - c_4^2 \sin^2\left(\frac{t}{R}\right)}{R^2} \right) dt \\ &= \int_{0}^{\tau} \frac{c_4^2}{R^2} \cos\left(\frac{2t}{R}\right) dt \\ &= \frac{c_4^2}{2R} \sin\left(\frac{2\tau}{R}\right). \end{aligned} $$

**Proof of Theorem 1.** Since $d_S(a, b) < \pi R$ the geodesic distance between $a$ and $b$ is not infinite there exists a path of finite length connecting $a$ and $b$ in $S$. If there is at least one path between $a$ and $b$ with finite length, then it is known that a minimizing geodesic, i.e. a path with minimal length connecting $a$ to $b$ exists (see the second paragraph of part III, section 1: “Die Existenz geodätischer Bogen in metrischen Räume” in [26]. Therefore the set we consider is not empty.

Let $\gamma_1, \gamma_2: [0, \ell] \to S$ be two length minimizing paths from $a$ to $b$, parametrized by arc length, with same length: $\text{length}(\gamma_1) = \text{length}(\gamma_2) = \ell := d_S(a, b)$.

Denote by $\epsilon$ the maximal distance between $\gamma_1(t)$ and $\gamma_2(t)$, i.e.

$$ \epsilon := \max_{t \in [0, \ell]} |\gamma_1(t) - \gamma_2(t)|, \qquad (5) $$

and let $t_{max} \in [0, \ell]$ be a parameter for this distance is attained, that is,

$$ d(\gamma_1(t_{max}), \gamma_2(t_{max})) = \epsilon. $$

<footnote n="3">³ Note that our the Lagrangian of a high dimensional harmonic oscillator (with constant potential) is the sum of one-dimensional harmonic oscillators.</footnote>
Without loss of generality, we assume that $t_{\max} \le \ell/2$.

We now consider the two families of curves

$$\gamma_{\lambda}^{E}(t) = \lambda\gamma_{1}(t) + (1 - \lambda)\gamma_{2}(t)$$

and

$$\gamma_{\lambda}^{S}(t) = \pi_{S}(\gamma_{\lambda}^{E}(t)).$$

We now upper bound the length $\gamma_{\lambda}^{S}(t)$. By Lemma 4 we have that

$$d(\gamma_{\lambda}^{E}(t), S) \le R - \sqrt{R^2 - \lambda(1 - \lambda)|p - q|^2}.$$

Using Corollary 5 this yields that

$$\text{length}(\gamma_{\lambda}^{S}(t))$$

$$\le \int_{0}^{\ell} \frac{1}{1 - (1 - \sqrt{1 - \lambda(1 - \lambda)|\gamma_{1}(t) - \gamma_{2}(t)|^2/R^2})} |\lambda\dot{\gamma}_{1}(t) + (1 - \lambda)\dot{\gamma}_{2}(t)| dt. \quad (6)$$

We now focus on the second term in the product in the integral.

With Lemma 6 we find the following bound on (6):

$$\text{length}(\gamma_{\lambda}^{S}(t)) \le \int_{0}^{\ell} \frac{1}{1 - (1 - \sqrt{1 - \lambda(1 - \lambda)|\gamma_{1}(t) - \gamma_{2}(t)|^2/R^2})} \cdot \sqrt{1 - (1 - \lambda)\lambda(|\dot{\gamma}_{1}(t) - \dot{\gamma}_{2}(t)|^2)} dt. \quad (7)$$

Assuming that $(1 - \lambda)\lambda$ is very small (more precisely we'll show that the length decreases in first order near $\lambda = 0$ or $\lambda = 1$) yields that

$$\begin{align*}
& \int_0^\ell \frac{1}{1 - (1 - \sqrt{1 - \lambda(1 - \lambda)|\gamma_1(t) - \gamma_2(t)|^2/R^2})} \\
& \qquad \cdot \sqrt{1 - (1 - \lambda)\lambda(|\dot{\gamma}_1(t) - \dot{\gamma}_2(t)|^2)} dt. \\
&= \int_0^\ell \left(1 + \lambda(1 - \lambda) \frac{|\gamma_1(t) - \gamma_2(t)|^2}{2R^2}\right) \left(1 - \frac{1}{2}(1 - \lambda)\lambda(|\dot{\gamma}_1(t) - \dot{\gamma}_2(t)|^2)\right) dt \\
& \qquad + \mathcal{O}(\lambda^2(1 - \lambda)^2) \tag{by Taylor} \\
&= \int_0^\ell \left(1 + \lambda(1 - \lambda) \frac{|\gamma_1(t) - \gamma_2(t)|^2}{2R^2} - \frac{1}{2}(1 - \lambda)\lambda(|\dot{\gamma}_1(t) - \dot{\gamma}_2(t)|^2)\right) dt \\
& \qquad + \mathcal{O}(\lambda^2(1 - \lambda)^2) \\
&= \int_0^\ell \left(1 - \frac{1}{2}(1 - \lambda)\lambda \left(|\dot{\gamma}_1(t) - \dot{\gamma}_2(t)|^2 - \frac{|\gamma_1(t) - \gamma_2(t)|^2}{R^2}\right)\right) dt \\
& \qquad + \mathcal{O}(\lambda^2(1 - \lambda)^2) \tag{8}
\end{align*}$$

Now writing $\Delta(t) = \gamma_1(t) - \gamma_2(t)$ and thus $\dot{\Delta}(t) = \dot{\gamma}_1(t) - \dot{\gamma}_2(t)$, this integral (up to some extra additional and multiplicative constants and ignoring the higher order terms) takes the form of a familiar Lagrangian integral or action

$$\int_{0}^{\ell} \mathcal{L} \, dt = \int_{0}^{\ell} \left( 1 - \frac{1}{2}(1 - \lambda)\lambda \left( |\dot{\Delta}(t)|^2 - \frac{|\Delta(t)|^2}{R^2} \right) \right) dt, \quad (9)$$
XX:8 Geodesics of length less than $\pi R$ in a set of reach $R$ are unique and continuous

which we recognize (up to global additive and multiplicative factors) as the action integral of the harmonic oscillator. To achieve better bounds we'll now split this into two integrals

$$ \int_{0}^{\ell} \mathcal{L} dt = \int_{0}^{\ell/2} \mathcal{L} dt + \int_{\ell/2}^{\ell} \mathcal{L} dt. $$

We'll only concentrate on the first integral, the second one can be treated in a completely symmetrical manner.

We now invoke Theorem 7 and Corollary 7, with $\tau = \ell/2$, together with (7), (8), and (9), to find that

$$ \begin{aligned} \text{length}(\gamma_{\lambda}^{S}(t)) &\le \int_{0}^{\ell} \left(1 - \frac{1}{2}(1-\lambda)\lambda\left(|\dot{\gamma}_{1}(t) - \dot{\gamma}_{2}(t)|^2 - \frac{|\gamma_{1}(t) - \gamma_{2}(t)|^2}{R^2}\right)\right) dt \\ &\quad + \mathcal{O}\left(\lambda^2(1-\lambda)^2\right) \\ &= \left(\int_{0}^{\ell/2} + \int_{\ell/2}^{\ell}\right) \left(1 - \frac{1}{2}(1-\lambda)\lambda\left(|\dot{\gamma}_{1}(t) - \dot{\gamma}_{2}(t)|^2 - \frac{|\gamma_{1}(t) - \gamma_{2}(t)|^2}{R^2}\right)\right) dt \\ &\quad + \mathcal{O}\left(\lambda^2(1-\lambda)^2\right) \\ &\le 2\left(\ell/2 - \frac{1}{2}(1-\lambda)\lambda \frac{c_4^2}{2R} \sin\left(\frac{\ell}{R}\right)\right), \\ &\qquad \text{(the } \le \text{ follows because } -\frac{1}{2}(1-\lambda)\lambda \text{ is negative)} \end{aligned} $$

where $c_4$ is some constant. This is strictly less than $\ell/2$, as long as $\ell < \pi R$. This contradicts the assumption that the geodesics have length $\ell$ (which is the sum of the integral from 0 to $\ell/2$ and the integral from $\ell/2$ to $\ell$), because now we found a curve that is shorter up to first order. ◀

## 2.3 Continuity

The uniqueness of the geodesics of length less than $\pi R$ implies continuity for geodesics under the same restriction. The proof of the continuity will rely heavily on the Arzelà-Ascoli theorem, see e.g. [12]. To this end we have to recall some nomenclature: A sequence $\{\gamma_n\}_{n \in \mathbb{Z}_{\ge 0}}$ of continuous functions on an interval $I \subset \mathbb{R}$ is uniformly bounded if for all $n$ and all $t \in I$, we have that $|\gamma_n(x)| \le C$, with $C$ some (universal) constant. The sequence $\{\gamma_n\}_{n \in \mathbb{Z}_{\ge 0}}$ is uniformly equicontinuous if, for every $\epsilon > 0$ there exists a $\delta > 0$, such that for all $t, t'$ satisfying $|t - t'| \le \delta$ and all $n$ we have that

$$ |\gamma_n(t) - \gamma_n(t')| < \epsilon. $$

The theorem can now be stated as

**Theorem 9 (Arzelà-Ascoli).** Let $\{\gamma_n\}_{n \in \mathbb{Z}_{\ge 0}}$ be a sequence of real-valued uniformly bounded, equicontinuous functions on a compact interval $I$. Then there exists a subsequence $\{\gamma_{n_k}\}_{k \in \mathbb{Z}_{\ge 0}}$ that converges uniformly.

With these preliminaries out of the way we can state the second contribution of the paper:

**Corollary 2.** Let $S$ be a set of positive reach $R$, then the geodesic connecting any two points $a, b \in S$ is continuous (in the $C^0$ sense) with respect to the end points provided that $d_S(a, b) < \pi R$.
Proof. Assume that the geodesics are not continuous with respect to the end points, then there exist an $\epsilon > 0$ and a pair of sequences of points $a_n, b_n$, where $\lim_{n \to \infty} a_n = a$ and $\lim_{n \to \infty} b_n = b$, as well as a sequence of geodesics $\gamma_n$, connecting $a_n$ and $b_n$ such that

$$ \sup_{t \in I} |\gamma_n(t) - \gamma(t)| \ge \epsilon. \qquad (10) $$

We stress that here we parametrize the geodesics on the unit interval $[0, 1]$.

Thanks to the metric characterization of sets of positive reach [6] we know that geodesics are themselves sets of (at least) the same reach as the set in which they lie. We also know that manifolds of positive reach are $C^{1,1}$, see [23], so that in particular geodesics are $C^{1,1}$, which makes them uniformly equicontinuous. The fact that the geodesics are bounded is straightforward, it can e.g. be seen as a consequence of the triangle inequality or of the metric characterization.

We now observe that by passing to a subsequence we can assume that there is one $t^* \in I$ such that

$$ |\gamma_n(t^*) - \gamma(t^*)| \ge \epsilon/2. \qquad (11) $$

This can be seen as follows: Because $I$ is compact the supremum must be attained at some point $t_n^*$ for each $(\gamma_n - \gamma)(t)$. Again because $I$ is compact this sequence of points must have a convergent subsequence $n_m$ with limit $t^*$ because of smoothness of $(\gamma_n)(t)$, we have that for $m$ sufficiently large and $|t - t^*|$ sufficiently small, we have $|\gamma_{n_m}(t) - \gamma(t)| \ge \epsilon/2$. This implies that (11) holds.

On the other hand the length of $\gamma_n$ has to converge to the geodesic distance between $a$ and $b$. If this were not true than the triangle inequality would yield a contradiction assumption that the curves involved are geodesics.

Now we can use the Arzelà-Ascoli theorem to conclude that there exist a $\gamma'$, which is the limit of the convergent subsequences $\gamma_n$ respectively, such that:

■ $\gamma$ and $\gamma'$ both are geodesics, that is their lengths equal the geodesic distance between the end points.

■ $|\gamma(t^*) - \gamma'(t^*)| \ge \epsilon/2$, in particular $\gamma \ne \gamma'$.

This yields a contradiction with the uniqueness of the geodesics, that is Theorem 1. ◀

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