The tangent cone, the dimension and the frontier of the medial axis

This paper establishes a relation between the tangent cone of the medial axis of X at a given point a∈Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathbb {R}}^n$$\end{document} and the medial axis of the set of points m(a) in X realising the Euclidean distance d(a, X). As a consequence, a lower bound for the dimension of the medial axis of X in terms of the dimension of the medial axis of m(a) is obtained. This formula appears to be the missing link to the full description of the medial axis’ dimension. An extended study of potentially troublesome points on the frontier of the medial axis is also provided, resulting in their characterisation by the recently introduced by Birbrair and Denkowski reaching radius whose definition we simplify.


Introduction
The medial axis, introduced by Blum in [1] as a central object in pattern recognition, emerges under various names in numerous mathematical and application problems. Lossless compression of data makes it an appealing object in tomography, robotics, or simulation. At the same time, its natural definition appears in various versions also in the fields of partial differential equations or convex analysis. Indeed, the medial axis is precisely the locus of points where the set distance function ceases to be differentiable. The distance function, on the other hand, is a viscosity solution for the most simple eikonal equation ∇d = 1 with the zero Dirichlet condition [2]. While it is a first-order equation, it is natural to wonder where the solution is C 1 -smooth. A deep connection with the initial set geometry makes the medial axis an interesting object for investigating geometrical and topological properties such as sets singularities [3] or homotopy groups [4]. The object also attains increasing attention in the context of Riemannian manifolds, where sets closely related to 27 Page 2 of 29 A. Białożyt NoDEA medial axes are studied under the name of cut loci [5][6][7]. However, extreme attention to detail is advised while studying the Riemannian generalisation as one has to take notice of non-Euclidean phenomena such as the existence of conjugate points or several geodesics with common endpoints, which stirs up the cut locus definition.
As medial axes are closely related to the extensively developed notion of conflict sets, one can believe that most of the theorems concerning conflict sets should have their counterparts in the medial axis theory. Unfortunately, the medial axes are (in)famous for their instability [8][9][10]. Thus the proofs are seldom transferable between the theories concerning these two objects. The main result of the present paper is a proof of the medial axis analogue of [11] Theorem 2.2. Since the proof presented by Birbrair and Siersma depends heavily on the monotonicity of Conflict Sets-a phenomenon with no counterpart in the medial axis setting-we are forced to develop an entirely new approach to the problem based on an analysis of the graph of the distance function. A similar problem was studied in a slightly broader sense and on the grounds of the convex analysis in [12]. Focusing precisely on the medial axis, we are able to provide more rigid results and formulae. An immediate application of the result answers the question about the dimension of a medial axis raised in [13,14]. Later in the paper, potentially troublesome points of the medial axis' frontier are characterised by a limiting directional reaching radius, an object combining the virtues of the Birbrair-Denkowski reaching radius [3] and Miura's radius of curvature [15]. It is adapted for the study of sets with higher codimension, and it also provides insights into the Birbrair-Denkowski archetype.
In this paper we restrict our attention to sets that are definable in the o-minimal structures expanding the field of real numbers. Such an approach gives us a framework with a handful of valuable tools. Firstly, the Hausdorff dimension of a definable set is always an integer and corresponds with the highest dimensional real vector space possible to map injectively into the set. Furthermore, the Curve Selection Lemma holds, meaning that every point in the closure of a definable set can be approached by a curve contained in the set. At the same time, the setting chosen protects us from pathological Cantorlike sets while conserving the applicability of the setting. Readers who are not familiar with the notion of definable sets may think of them as semialgebraic sets. An excellent introduction to the notion is found in [16,17]. For a closed, nonempty subset X of R n endowed with the Euclidean norm, we define the distance of a point a ∈ R n to X by which allows us to introduce the set of closest points in X to a as We will usually drop the indices of the (multi-)functions d and m.
The main object discussed in this paper is the medial axis of X denoted by M X , that is, the set of points of R n admitting more than one closest point in the set X, namely A descriptive way, the most often invoked, to imagine the medial axis, brings an image of the propagation of a fire front starting at X. In this case, the medial axis of X is precisely the set of points where fronts originating from different starting points meet. This picturesque idea illustrates maybe the most profound feature of the medial axis-it collects exactly those points of the ambient space, at which the distance function is not differentiable (Fig. 1).
As an introductory remark, it is worth recalling that, as was shown in [13], both the medial axis and the multifunction m(x) are definable in the same structure as X. Moreover, the multifunction m(x) is upper (outer) semicontinuous, meaning for any set A with a 0 in its closure.

The tangent cone of the medial axis
Let us begin by recalling that we have an explicit formula for the directional derivative of the distance function due to Richard von Mises [20] (most often misquoted as M. R. de Mises, the first 'M.' clearly standing for Monsieur, see also [21]). Theorem 2.1. (R. von Mises) Let X be a closed, nonempty subset of R n , then for every point a ∈ R n \X all one-sided directional derivatives of the distance function d X exist and are equal to Proof. For any a, b ∈ R n , there is a, b = a b cos α, where α denotes the angle between a and b. Thus, for v = 1 the assertion can be written as where α x is the angle between x − a and v. Clearly, the value of − cos α will be the smallest for the smallest α. Without loss of generality, assume v = (1, 0, . . . , 0) and a = 0, then take x 0 ∈ m(0) forming the smallest angle with v, and Denote now by α t the angle formed by v and x t . By the Cosinus Theorem applied to the triangle formed by tv, 0, x t , we have Keeping in mind d(0) = x 0 we can clearly see that Both d(tv) and x t converge to d(0) as t → 0, so the proof will be completed if only α t → α 0 and xt 2 − x0 2 t → 0. Actually, both claims can be derived from the closedness of X. Indeed, closedness guarantees that for any ε > 0, we can find such δ > 0 that for all x ∈ X with the first coordinate greater than or equal to x 0 ≤ ε as long as x < d(0) + δ, otherwise x 0 would not realise the smallest angle among those formed by v and x ∈ m(0). Assuming that m(0) is a subset of the unit sphere, we can use the polarization identity for the inner product to express D v d(0, X) in the convenient form Since the infimum is attained at y ∈ m(0), which is closest to v, the formula simplifies even further down to The appearance of d m(0) in the formula for D v d X (0) brings along interesting consequences and possibilities to describe the medial axis' cone within its category. However, our first result will be independent of the Mises Theorem, for the situation for the subsets of the sphere is more straightforward than the general one (Fig. 2).
In what follows C a E denotes the classical Peano tangent cone of a set at its accumulation point a, i.e. the cone formed by all the directions v = Proof. Start by observing that since Y is a subset of S and an open ball in the induced metric is the intersection of a ball in R n with S, clearly M S Y ⊂ M Y . To finish this part of the proof, we need to show that M Y is a cone. This follows from the observation that the intersection of S and any ball B(x, d(x, m(x))), is always equal to a closed ball in S centered at x/ x with radius d(x/ x , m(x)) = d(x/ x , m(x/ x ).
The second part of the theorem is trivial, as M Y is a cone, and every vector λv, where λ ∈ R + , v ∈ M Y , approximating an element of a tangent cone C 0 M Y belongs to M Y .

A. Białożyt
NoDEA Even this simple situation demands some finesse. On the one hand, it is indeed necessary to take the closure of M Y in Proposition 2.3. As seen for Y = {xy = 0} ∩ S ⊂ R 3 , a point v = (0, 0, 1) lies in Y so it cannot belong to M Y . Nevertheless, it is easy to check that v is a point of the tangent cone at the origin of the medial axis. Additionally, since closed balls both in the spherical and in the induced metric are the same, the choice of the metric used to calculate M S Y in Proposition 2.3 does not affect the assertion. We will use the theorem of the Denkowskis [19].
Remark 2.5. One of the most important corollaries of Theorem 2.4 binds the tangent cone of the medial axis with the medial axis of a tangent cone for definable sets. Namely, in the definable setting, due to Curve Selection Lemma, the tangent cone C 0 X of a set X ⊂ R n at a point X = 0 is given as the Kuratowski limit of the set dilatations Since scaling the set scales its medial axis accordingly, Theorem 2.4 asserts that Even though the cone inclusion from the last remark looks promising to describe the tangent cone of a medial axis at an arbitrary point, one stumbles upon a gnawing obstacle while applying it to the problem. Since points of a medial axis M X are separated from the set X, the lefthand side of the inclusion from Remark 2.5 becomes empty, and the approach fails to deliver any meaningful insight. Thus, to tackle the problem of the description of the tangent cone, we need to be a trifle more cunning. In fact, we require of a technical lemma which describes the geometry of the graph of the distance function.
Lemma 2.6. For any closed X ⊂ R n , a graph has the following properties:

in other words: the medial axis of the epigraph of d is equal to
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Proof. 1. is a consequence of the Lipschitz condition for the distance function.

comes from m(tv
3. can be proved by observing that 1. and 2. give together: (2) gives (x, y) ∈ Γ, moreover for a certain t ∈ [0, 1], there is Thus, It is easy to check now that for every point p of the axis of the cone where p (n+1) < d(a) denotes the last coordinate of p.
With the properties of the graph of the distance function at hand, we are ready to prove: Proof. If 0 ∈ M X \M X , the theorem is trivial as M m(0) = ∅. Assume then, without loss of generality, that 0 ∈ M X and d(0) = 1. Denote by Γ the graph of the distance function d as was done in the previous lemma. According to Remark 2.5, M C (0,1) Γ ⊂ C (0,1) M Γ . To prove the theorem, we will establish the relation between these sets and M m(0) and C 0 M X .
Let us begin with C (0,1) M Γ . Since Lemma 2.6(3) gives the tangent cones of M Γ − (0, 1) and M X × R must coincide in the cone {y ≤ α x } for any choice of α < −1. Because R is a cone, we further obtain the coincidence of C (0,1) M Γ and C 0 M X × R in the aforementioned cone.
As it comes to M m(0) and M C (0,1) Γ , we will investigate first the set C (0,1) Γ. Since d X is a Lipschitz function, the explicit formula for the directional derivative D x d X (0) allows us to express C (0,1) Γ as the graph of the function Consider for a moment the graph Γ 1 of the function x → d(x, m(0)). For x < 1, it has the structure of a cone with a vertex at (0, 1), furthermore the tangent cone C (0,1) Γ 1 can be expressed as the graph of the same function as in the case of Γ, namely x → D x d X (0). The medial axis of the epigraph of d(x, m(0)) after the translation by (0, −1) has to coincide with M C (0,1) Γ , thus their intersections with the cone {y ≤ α x } are also equal.
As the following example shows, the equality between C a M X and M m(a) cannot be expected in general.
It is easy to check that indeed M m(0) is a proper subset of C 0 M X (Fig. 3).
The reconstruction of the whole tangent cone of M X based solely on m(a) may not be possible due to sequences of points x ν in the medial axis with m(x ν ) converging to a singleton. Assuming no such sequence can be found, we can prove the following. Proof. Theorem 2.7 gives us one of the inclusions in question. To prove the other one, start by taking v ∈ C 0 M X . By definition, we can find sequences Take any convergent sequence of elements m(a ν ) x ν → x ∈ m(0); we will show that x ∈ m m(0) (v). Since the additional assumption on the diameter of m(a ν ) ensures diam lim sup m(a ν ) ≥ r > 0 this will . Since x ν → x, the sequence of balls B ν converges to some closed ball B centered at v with x on its boundary. Of course, the interior of B has an empty intersection with m(0), as for every The author suspects, that the equality holds everywhere outside of M X \M X . However the proof does not seem to be simple.
As for the parts of the tangent cone at a point x of the medial axis which are not detected by m(x), it is still possible to distinguish parts of X contributing to C x M X . While verifying whether v ∈ S belongs to the tangent cone of M X , the investigation of X can be restricted to a neighbourhood of m m(0) (v). A. Białożyt NoDEA Proposition 2.9. Assume that 0 ∈ M X . For any v ∈ S and δ > 0, there exist r, ε > 0 such that for Proof. Without loss of generality we can assume that d(0, we will show first that for an arbitrary ζ > 0 we can find r, ε > 0 such that Observe firstly that for a ∈ R n and y ∈ m m(0) (v) an inequality holds d(a, X) ≤ d(a, y) = 1 + a 2 − 2 a cos ∠(a, y).
Since the angle ∠(v, y) is strictly smaller than π, we can choose a positive ε 0 < π − ∠(v, y). Then for a ∈ R n forming an angle ∠(a, v) < ε 0 we have Now for any η > 0 we can shrink the ε 0 down to ε η to obtain and then pick r η > 0 such that a < r η implies Therefore it is possible to estimate d(a, X) < 1 − a (cos ∠(v, y) − 2η) =: R a with η arbitrary close to zero for a ∈ C(v, r, ε) with r < r η , ε < ε η .
To end the first part of the proof we need to show that for any a ∈ C(v, r, ε), we have B(a, R a )\intB(0, 1) ⊂ b(v) ζ . First observe that by shrinking r and η we can obtain B(a, R a ) ⊂ B(0, 1 + ζ/2). Now we need to prove that m m(0) (v))+ ζ/2 and can be verified in the plane spanned by v and a. However, the inequality holds uniformly for η, r, ε small enough in any such plane, due to Pythagorean Theorem. Now, since X is closed and m(0) ∩ b(v) = m m(0) (v), for any δ > 0 we can find a positive number ζ such that Since all points in C(v, r, ε) have their distance to X realised in b(v) ζ , the assertion follows. Proof. From Theorem 2.7 the medial axis M m(a) is a subset of the tangent cone C a M X . Thus, its dimension is bounded by dim C a M X . Since M X is definable, dim M X is always greater than or equal to dim C a M X , and the assertion follows.
The following result was known before (e.g. [12] Theorem 6.2 or [13] Theorem 4.10 for the subset of points in the medial axis with dim m(a) = n−1), Theorem 2.7 yields a proof that is more natural in the medial axis category. Proof. The sufficiency of the condition is apparent. For the proof of the necessity, suppose that m(a) does not fill the sphere S(a, d(a)) entirely. It is easy to observe (for example, using the compactness of the sphere, the continuity of the distance function, and Theorem 2.3), that its medial axis is a cone of dimension dim M m(a) > 0. From Theorem 2.7 we derive that dim C a M X > 0, hence a cannot be isolated.
In other words, every eyelet in m(a) enables an escape of M X in its general direction.
In the plane, the situation is, as usual, more straightforward.
Theorem 2.12. For a closed definable X ⊂ R 2 , there is always Proof. Assume a = 0. Of course, given Theorem 2.7, the only strict inclusion possible is M m(0) C 0 M X . Take then v ∈ C 0 M X \M m(0) if such a point exists.
Since M X is definable, by the Curve Selection Lemma, there exists a continuous γ : [0, 1] → M X with γ(0) = 0, tangential to v and such that 0 / ∈ γ((0, 1]). Moreover, as for every x ∈ m(0) a segment (0, x] does not intersect M X , the image of γ and the set (0, 1] · m(0) must be disjoint. Take now ν ∈ M m(0) ⊂ C 0 M X lying in the same connected component of R 2 \(R + · m(0)) as v and denote by ψ a curve in M X tangential to ν. For a vector w ∈ R 2 , there is lim t→0 + tw = 0 and lim sup t→0 + m(tw) ⊂ m(0). Therefore, for any w in a segment (v, ν) with an arbitrary choice of w t ∈ m(tw), starting from a certain T > 0, a segment [tw, w t ] has to intersect eventually either γ or ψ, which ends in a contradiction.  Figure 4. Theorem 2.12. Starting from a certain T > 0, the segment [tw, w t ] intersects either γ or ψ A slight variation of the Theorem 2.12 proof shows that the planar medial axes cannot form cusps. This is a result analogous to the no cusp property of the conflict sets from [11]. Corollary 2.14. For a closed definable X ⊂ R 2 and a ∈ M X , the germ of M X at a is a union of definable curves γ 1 . . . γ s with distinct tangent cones at a.
Proof. Assume, for sake of contradiction, that two distinct curves γ 1 , γ 2 in the decomposition of the germ of M X share the same tangent cone. The curves γ 1 , γ 2 are disjoint for t positive and small enough due to definability. Therefore, we can find a curve ϕ originating from zero and entirely included in the region bounded by γ 1 , γ 2 . Then, for t small enough and v t ∈ m(ϕ(t)), we obtain an analogous contradictory intersection of [ϕ(t), v t ] with one of the initial curves. Then M X = A, thus it is a pure two dimensional set whilst C 0 M X = {(t, 0, 0) | t ≥ 0}.

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Remark 2. 16. The results in this section bear the same flavour as those provided by Albano and Cannarsa in [23], however, concentrating on the distance function rather than a general semi-concave function, we made our approach to the tangent cone more comprehensive. The simple lower bound estimate of Corollary 2.10 is slightly sharper than one obtained by Albano and Cannarsa.
For an example of two circles of equal radius equidistant from the origin of R 3 , our method bounds the dimension of the medial axis by 2, whereas methods described in the cited paper allow to bound it only by 1. Furthermore, focusing on the distance function allowed us to derive the sufficient condition for the equality of the tangent cone and the medial axis of the closest points and several other plane-related results such as no cusp property.

The dimension of the medial axis
Our aim now is to use the established relation between the tangent cone of the medial axis at a point x ∈ M X and the medial axis of m(x) to describe the dimension of the medial axis in a more refined way than Corollary 2.10. This result can be viewed as an answer to the conjecture posted in [13,14]. We will use a notion of a cylindrical definable cell decomposition (cdcd ) from the o-minimal geometry, cf. [16]; it serves the role of quasi-stratification of a definable set.
or a definable band between two such functions Without much effort and implications on what follows, functions defining cells can be assumed to be of class C k for an arbitrary k ∈ N. However, in this paper we will not need such smoothness, thus just the continuity of the functions is assumed.

Definition 3.2.
We call a family of definable cells C a cylindrical definable cell decomposition (cdcd) of R n if the family C is finite, the cells of C are pairwise disjoint, C = R n , and a collection of projections on the first (n − 1) coordinates {π(C), C ∈ C} is a cdcd of R n−1 .
We say that a cdcd C is adapted to a finite collection of definable sets A 1 , . . . , A k , if A i is the union of certain cells of C for all i ∈ {1, . . . , k}. It is a standard result from o-minimal geometry that such adapted cdcd exists for any finite collection of definable sets [16]. Adapted cdcd serves to some extent the role of stratification known from semialgebraic or analytic geometry in the 27 Page 14 of 29

A. Białożyt
NoDEA definable setting. Since every cell C of a cdcd is homeomorphic to a finitedimensional real vector space, it follows that the dimension of a definable set A is equal to dim A = max{dim C | C ⊂ A} for a cdcd adapted to A, which does not depend on the choice of a cdcd. The local dimension of A at a point a ∈ A is defined then as Recall again that the multifunction m(x) is definable if X is a definable set. By definition it means that the graph is definable and thus there exists a cdcd {D 1 , . . . , D α } of R n × R n adapted to the intersection Γ m ∩ (M X × R n ). From the definition of cdcd, the collection of the projections of D i onto the first n coordinates forms a cdcd of R n adapted to M X . With this decomposition, we will prove further properties of the multifunction m(x), and as a consequence, we will obtain an explicit formula for the dimension of M X .

Definition 3.3.
Let D be a cdcd of R n × R n adapted to Γ m . Denote by C the cdcd of R n obtained from the projections of the cells in D onto the first n coordinates. We call Proof. It is known, cf. [3,13], that the multifunction m(x) is upper semicontinuous along M X : for any x 0 ∈ R n , we have the inclusion lim sup MX x→x0 m(x) ⊂ m(x 0 ). To prove the continuity of m| MX , we need to show that m(x 0 ) is a subset of the lower Kuratowski limit. Explicitly, we need to show that where U, V are open sets.
Take y = (y 1 , . . . , y n ) ∈ m(x 0 ) and a neighbourhood U 1 × . . . × U n of y. Denote by D the cell in D containing (x 0 , y) and by C its projection on R n . We will show the continuity of (π i • m)(x) where π i is the natural projection on the first i coordinates. Of course, (π n • m)(x) = m(x) and the assertion will follow.
For the first coordinate, observe that C 1 := (id R n × π 1 )(D) is either a graph of a continuous function or a band between two such functions defined over C. In the first case, we can easily find a neighbourhood V of x 0 in M X such that (π 1 • m)(V ) ⊂ U 1 which implies (π 1 • m)(V ) ∩ U 1 = ∅. In the case C 1

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The tangent cone, the dimension and the frontier... Page 15 of 29 27 is a band between two functions f − , f + , there must be f − (x) < y 1 < f + (x). By continuity these inequalities must hold in a certain neighbourhood V of x 0 , resulting in Assume now the composition π k • m to be continuous for k < n. Again, the cell C k+1 := (id R n × π k+1 )(D) is either a graph of a continuous function f k+1 or a band between two such functions, this time defined over C k . In the first case, we can find a neighbourhood V = V 0 × . . . × V k of (x, y 1 , . . . , y k ) such that f k+1 (V ) ⊂ U k+1 . As the composition π k •m(x) is continuous, we can ensure (x, y 1 , . . . , y k ) ∈ V 0 × V 1 × . . . × V k just by shrinking V 0 ; by shrinking it even further, we can also ensure that (y 1 , . . . , The case of C k+1 being a band follows in the same manner.
As was mentioned earlier, the inclusion results in the continuity of m| MX at every point of C, in particular at x 0 .
The proof of Proposition 3.5 gives a soothing correlated result on the continuity of m(x). In the o-minimal setting, the set of points at which m(x) is discontinuous is always nowhere dense and is equal to the medial axis (see [24] for a counter-example outside of the o-minimal setting). Luckily enough, the restriction of m(x) to the medial axis exhibits an analogous behaviour. It is still continuous outside of a subset nowhere dense in the induced topology. Consequently, with R n being locally compact, we have The main result of this paper, settling the question about the dimension of the medial axis in the definable case, is the following.

t. D, then an equality occurs
Proof. Without loss of generality, we can assume x 0 = 0.
We will prove the theorem by induction on the dimension of C-the cell containing x 0 . The case dim C = 0 is already proved in Corollary 2.11. Assume now that the dimension of C equals k, and the theorem holds for any cell C of dimension smaller than k. Since 0 satisfies the assumptions of Proposition 3.5, we can think about C 0 M X as the medial axis of m(0), then clearly Corollary 3.6 states that we can find 0 = v ∈ M m(0) for which the tangent cone to C 0 M X at v depends only on points in m(0) that are closest to v (and are collected in a set m m(0) (v)). Therefore, for such v, after a suitable translation, there is again The set m m(0) (v) is a subset of both S(0, d X (0)) and S(v, d m(0) (v)). Denote by L the unique n − 1 dimensional affine subspace of R n containing the intersection of the mentioned spheres. It is immediate that the subspace L can be written as v ⊥ + αv for a certain α ∈ R and that m m(0) (v) is a subset of L. The opposite inequality is far less complicated in proof. It suffices to observe that U ∩ C = U ∩ M X is a warrant that the dimension of m(x) is constant in U . Consequently Theorem 4.13 from [13] ensures that the sum dim 0 M X + dim m(0) cannot exceed n − 1.
Surprisingly, the formula for a generic point is enough to describe the dimension at any point of M X . This strengthening of the results from [12,13], and [25] solves the problem for sets definable in the o-minimal setting.
Proof. We will prove that min{k | a ∈ M k } = n − 1 − α ⇐⇒ dim a M X = α holds for any α ∈ N.
For α = 0, one of the implications is precisely the statement of Corollary 2.11. The opposite one is given by Theorem 4.10 from [13]. Now assume the claim to be valid whenever α < α 0 . Mind that Theorem 4.13 from [13] states that for any It remains to prove the opposite implication. Take any a ∈ M X with dim a M X = α 0 . The claim for α < α 0 allows us to analyse just the points in the vicinity of a where the local dimension of M X equals α 0 . Furthermore, only dim m(a) ≥ n − 1 − α 0 needs to be shown. Finally, we give a global formula for the dimension of the medial axis (Fig. 6). In particular, Proof. Obvious from Theorem 3.8.
Remark 3.10. If X is a collection of isolated points, its medial axis M X is exactly the conflict set of the singletons included in X (also called the Voronoi diagram of X). The dimension of m(a) equals zero for any point a ∈ R n , and the global formula indeed gives dim M X = n − 1, as predicted by the theory of conflict sets [11]. Conversely the conflict set theory assures that the dimension dim a M X = n − 1, whenever for a given a ∈ M X the set m(a) is not connected. The general formula for the medial axis dimension concludes that in every neighbourhood of a, a point b of M X must exist with m(b) finite.

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Remark 3.11. The results of this section correspond to ones brought by Ambrosio, Cannarsa and Soner in [12] for singularities of semi-convex functions. Their methods concentrated on investigating the set S k := {x ∈ M X | dim conv(m(x)) = k}. Localising the study in R n , they obtained the dimension formula dim S k = n−k. In general, no easy transition between S k and M k exists. However, comparing the local dimension formulas, we can derive that for any a ∈ M X , we necessarily have min{k | a ∈ S k } = min{k | a ∈ M k }+1. Furthermore, for a generic point a ∈ M X , we have min{k | a ∈ S k } = dim m(a) + 1.

The frontier of the medial axis
In [15] T.Miura proposed a characterisation of the medial axis boundary for hypersurfaces. Unfortunately, the introduced notion did not escape flaws. Foremost, it does not recognise the studied side of the hypersurface, which may result in misleading data. In this paper, we provide an improved definition, resulting in a generalisation of the claims from [15], proved with more straightforward reasoning.
To take notice of the direction of open balls used to investigate X ⊂ R n , we assume the following definitions. Take any point a ∈ X and write to be the normal cone to X at a. Then denote the set of directions normal to X at a by V a := N a X ∩ S and limiting set of normal directions by Remark 4.1. If a ∈ X is a point of C 1 -smoothness, then, of course, the tangent spaces, and, what follows, the normal spaces are continuous at a. Therefore, the limiting set of normal directions is just the set of normal directions in such a case.
Using the introduced sets, for a point a ∈ X we consider the following definition.

Definition 4.2.
For v ∈ V a , we define a directional reaching radius by Then for v ∈ V a we define a limiting directional reaching radius bỹ Recall that for any v ∈ N a X, the point a + vr v (a) is the centre of a maximal (in the sense of inclusion) ball contained in R n \X. The set of centres of maximal balls are gathered in a set called the central set of X. It is known 27 Page 20 of 29 A. Białożyt NoDEA (cf. [3]) that the central set of a closed set lies between the medial axis of the set and its closure.
At first glance, the infimum in the definition of the reaching radius might seem dubious for points with an empty limiting set of normal directions. By definition of the infimum, we are inclined to posit an infinite value of the reaching radius for any such point. Luckily, in the o-minimal geometry, the points with an empty limiting set of normal directions prove to be precisely the interior points of a given set. Clearly, any interior point has an empty limiting set of normal directions. Conversely, every boundary point of a definable set can be reached by the regular part of the boundary. This is a consequence of the definability of the boundary and the nowhere density of the subset of singularities. Points of the regular part of the boundary of X have at least one normal direction. Thus, due to the compactness of the sphere, the limiting set of normal directions for the boundary points cannot be empty.

Remark 4.3.
As is easily seen from their definitions, bothr v (a) and r(a) are lower semi-continuous functions. Moreover, for v ∈ V a \V a the limiting directional reaching radius equals zero.
Recall that Reg k X denotes the set of points of X at which it is a C ksmooth submanifold. Accordingly, the C k -singularities are denoted by Sng k X = X \ Reg k X.
The backbone of the just defined reaching radius lies in the same place as the reaching radius introduced in [ is called weak reaching radius. One can perceive the difference between them as a sort of an order of taking limits problem. This paper emphasises the medial axis and a point directional arrangement, whereas Birbrair and Denkowski focused on their distance. Since the Birbrair-Denkowski reaching radius proved to be successful in describing M X ∩ X, it would be desirable to achieve at least a type of correspondence between these two notions. Fortunately, as we will see in Theorem 4.13, the final results of both constructions are equal. For the sake of the next preparatory proposition, recall that the normal set at a ∈ X defined in [3] as N (a) := {x ∈ R | a ∈ m(x)} is always convex and closed. Moreover, for a subset of a unit sphere A denote by conv S (A) its convex hull in the spherical norm. is upper semi-continuous on V a . Furthermore, it is continuous at v ∈ V a , if there exist r, ε > 0 such that B(v, r) ∩ V a = conv S (S(v, r) ∩ V a ) and v / ∈ ρ −1 ([0, ε]).
Proof. Observe first that for an empty V a , the function ρ is continuous by definition. Therefore, assume for the rest of the proof that V a is nonempty.
To prove the upper semi-continuity take v 0 ∈ V a and any sequence of v ν → v 0 . Then, for any r < ρ(v ν ) the point (a + rv ν ) lies in N (a). Therefore, from the closedness of N (a), the point (a + rv) must lie in N (a) for every r < lim sup v→v0 ρ(v) . Moreover, N (a) is convex, so the whole segment [a, a + rv 0 ] must be a subset of N (a) as well. This inclusion means that ρ(v 0 ) ≥ lim sup v→v0 ρ(v).
Mind that even though the normal set N (a) is convex for any a ∈ X, it does not necessarily mean that an r > 0 satisfying B(v, r)∩V a = conv S (S(v, r)∩ V a ) for every v ∈ V a exists. Indeed, only an inclusion from right to left is automatic. Take, for example, X = {z = x 2 + y 2 }. We can see that V 0 = {z ≤ − x 2 + y 2 } ∩ S, thus for v = (1, 0, −1) and all r > 0, there is B(v, r) ∩ V 0 conv S (S(v, r) ∩ V a ).