Regularity of distance functions from arbitrary closed sets

We investigate the distance function $\boldsymbol{\delta}_{K}^{\phi}$ from an arbitrary closed subset $ K $ of a~finite-dimensional Banach space $ (\mathbf{R}^{n}, \phi) $, equipped with a uniformly convex $\mathcal{C}^{2}$-norm $ \phi $. These spaces are known as \emph{Minkowski spaces} and they are one of the fundamental spaces of Finslerian geometry (see https://doi.org/10.1016/S0723-0869(01)80025-6). We prove that the gradient of $\boldsymbol{\delta}_{K}^{\phi}$ satisfies a Lipschitz property on the complement of the $\phi$-cut-locus of $K$ (a.k.a. the medial axis of $\mathbf{R}^{n} \sim K$) and we prove a~structural result for the set of points outside $K$ where $\boldsymbol{\delta}_{K}^{\phi}$ is pointwise twice differentiable, providing an answer to a question raised by Hiriart-Urruty (see https://doi.org/10.2307/2321379). Our results give sharp generalisations of some classical results in the theory of distance functions and they are motivated by critical low-regularity examples for which the available results gives no meaningful or very restricted informations. The results of this paper find natural applications in the theory of partial differential equations and in convex geometry.


Introduction
For the basic notation we refer the reader to section 2.1.
Suppose K ⊆ R n is a closed set and φ is a uniformly convex norm on R n ; cf. 2.8. Our central object of study is the φ-distance function We investigate in detail the set of points where δ φ K is not differentiable and then also the set of points where it is not pointwise twice differentiable. Define (1) Σ φ (K) = (R n ∼ K) ∩ x : δ φ K is not differentiable at x .
A basic and fundamental result in the theory of distance functions asserts what follows.
1.1 Theorem (C 1,1 -regularity). If K ⊆ R n is an arbitrary closed set, then δ φ K is C 1 with a locally Lipschitz gradient on the open subset U := R n ∼ K ∪ Clos Σ φ (K) .
This result can be deduced employing general results from the theory of Hamilton-Jacobi equations (see [Lio82,Theorem 15.1] or [Fat03]). Indeed, for a general closed set K it is well known that δ φ K is a locally semiconcave function on R n ∼ K and it satisfies, in a viscosity sense, the Eikonal equation φ * (grad u) = 1 on R n ∼ K (where φ * is the dual norm of φ as defined in 2.7); see [Asp73], [Lio82], [Zaj83a]. For the Euclidean norm Theorem 1.1 can also be obtained using a purely geometric argument (see [Fed59,4.8]).
Of course, the conclusion of the theorem can be improved if we know that K is at least a C 2submanifold. In fact, in this case δ φ K is at least of class C 2 on the open subset U and Clos Σ φ (K) is a set of L n -measure zero; if K is a C 2,1 -submanifold, then Clos Σ φ (K) is a set of locally finite H n−1 -measure; see [IT01], [MM03], [LN05a], [LN05b], and [CM07]. A sufficient condition that In particular, for H n−1 almost all (a, η) ∈ N φ (K) the distance function δ φ K is pointwise twice differentiable at all points of the line segment {a + rη : 0 < r < r φ K (a, η)}.
The proof of Theorem 1.5 is based on the Lipschitz property proved in Theorem 1.4 and on some general estimates for the pointwise principal curvatures of level sets of δ φ K (these level sets might not even be topological manifolds but they admit a natural notion of pointwise curvature; see 2.44). Moreover, results on the preservation of the density points under bilipschitz transformations (see [Buc92]) and on the approximate differentiability of multivalued functions (see 2.38) are used in a crucial way.

Applications
Here we briefly mention a couple of different applications of the results of the present paper.
Pointwise regularity and gradient Lipschitz estimates for solutions of the Eikonal equation. Suppose Ω ⊆ R n is an arbitrary open set, φ is a uniformly convex C 2 -norm and φ * (u) = sup{u • v : φ(v) = 1}. It is well known that δ φ K is the unique viscosity solution of the following Eikonal equation on Ω φ * (grad u) = 1 on Ω u = 0 on ∂Ω If ∂Ω is hypersurface of class at least C 2 , then the local structure of this solution has been extensively studied and it is by now very well understood (see the references cited at the beginning of this introduction). On the other hand, as already explained, if ∂Ω is not C 2 then such a solution can have a very complicated (in particular dense!) singular set (see [San21]) and many classical results in the theory do not give an insight about its local structure. In this direction our results in Theorems 1.4 and 1.5 provide a new and rather sharp description of the structure of the solutions of the Eikonal equation for arbitrary domains.
Steiner formula and curvature measures in uniformly convex finite dimensional Banach spaces One of the original motivation of the second author for the present work is to provide results that can be used to advance the theory of convex and integral geometry in Minkowski spaces; see [Hug99]. In [HS22] the second author in collaboration with Daniel Hug employs Theorems 1.4 and 1.5 to prove the Steiner formula for arbitrary closed sets in a uniformly convex Banach space (Minkowski space); thus, extending the same formula previously obtained in [HLW04] in the Euclidean space. The Steiner formula is then used as a starting point to develop the theory of curvature measures for sets of positive reach in a Minkowski space.

Notation
We follow traditional well established and widely accepted conventions and notations typical for geometric measure theory. For convenience of the reader we briefly describe them here. We use the following symbols L n the Lebesgue measure over R n ; α(k) Lebesgue measure of the unit ball in R k ; S n−1 the unit Euclidean sphere in R n ; x • y the inner product of two vectors x and y in a Euclidean space; |x| the norm of a vector x in a normed vectorspace; A ∼ B set-theoretic difference of two sets A and B; Clos A closure of a subset A of a topological space; Int A interior of a subset A of a topological space; Hom(X, Y ) vectorspace of linear maps of type X → Y ; Λx or x , Λ the value of a linear map Λ on a vector x ∈ dmn Λ; ≤ r} closed ball with respect to a norm φ; f |A restriction of a function f to the set A ⊆ dmn f ; ∇f (x) the set of subgradients of a convex function f at x ∈ dmn f ; cf. 2.20 and 2.21; T the linear orthogonal projection onto a linear subspace T of a Euclidean space; T ⊥ the orthogonal complement of a linear subspace T of a Euclidean space; [A x → f (x)] an unnamed function defined on A whose value at x ∈ A is f (x); Given k ∈ Z + and 0 < α < 1 we shall say that a function f is of class C k,α if the k th derivative D k f exists and satisfies the Hölder condition with exponent α; cf. [Fed69, 3.1.11 and 5.2.1]. We say that f is of class C k if D k f is just continuous.
2.1 Remark. We study several notions depending on the norm φ, whose name is always in the superscript. In case φ is the standard Euclidean norm on R n we omit it in the notation so, e.g., if x ∈ R n and 0 < r < ∞, then U(x, r) denotes an open Euclidean ball in R n .
We now introduce some classical functions • Hausdorff densities of a Radon measure µ at x and Θ n (µ, x) = Θ * n (µ, x) whenever Θ * n (µ, x) = Θ n * (µ, x) ; • dilations µ r (x) = rx whenever r ∈ R and x is a vector ; • translations τ a (b) = a + b whenever a and b are vectors in a vectorspace X ; • the identity map on a set X I X (x) = x whenever x ∈ X .
2.2 Remark. Without introducing any new symbols (in order not to make the notation too heavy) we find that given a function f defined on a subset of a normed vectorspace dmn Df is the set of differentiability points of f .
2.3 Remark. We shall repeatedly make use of the following simple fact. If f is a real valued function defined on a subset of a Euclidean space X, x ∈ dmn D 2 f , and u, v ∈ X, then 2.4 Remark. We adopt the convention that "C x.y (a, b, c)" refers to the object (e.g. constant) defined in item (lemma, theorem, corollary, remark) x.y under the name "C", where a, b, c should be substituted for parameters of x.y in order of their occurrence. For instance, if v is a vector such that φ(v) = 1, then M 3.7 ( 1 2 , v) is the manifold constructed by employing 3.7 with 1 2 and v in place of "ε" and "η".

Basic concepts
2.5 Definition. We say that a norm φ : 2.6 Remark. In the sequel, unless otherwise specified, n shall be a fixed positive integer, X will be a vectorspace of dimension n, and φ : X → R will be a strictly convex norm on X of class C 2 away from the origin. Of course, X shall be isomorphic with R n but, whenever we write X instead of R n , we want to emphasise that there might not be a natural choice of a Euclidean structure on X.
2.7 Definition. Whenever X is equipped with a scalar product and φ : X → R is a norm we define the conjugate norm φ * : X → R by the formula φ * (x) = sup x • y : y ∈ X , φ(y) = 1 for x ∈ X .
2.8 Definition (cf. [DRKS20, 2.12, 2.13]). Assume X is equipped with a Euclidean structure. We say that φ : X → R is a uniformly convex norm if it is a norm and there exists γ > 0 such that the function X x → φ(x) − γ|x| is convex.
2.10 Definition. Given a closed set K ⊆ X we define 2.11 Definition. A map of the type f : X → 2 Y shall be called Y -multivalued. In case x ∈ X and f (x) is a singleton, we abuse the notation and write f (x) to denote the unique member of f (x).

2.12
Definition. Let f be a Y -multivalued function on X and A ⊆ X. Then we denote with f |A the Y -multivalued map on X defined as 2.13 Definition. Let f be a Y -multivalued function on X and A ⊆ X. Then we define the inverse f −1 of f as the X-multivalued map on Y as 2.14 Definition. Suppose K ⊆ X is closed and ξ φ K : X → 2 K is the φ-nearest point projection onto K characterised by (2). The Cahn-Hoffman map of K associated to φ is the multivalued map ν φ K : X ∼ K → 2 ∂B φ (0,1) defined by the formula 2.15 Remark. It will be useful to notice that ξ φ K (x) is a compact subset of X for every x ∈ X. 2.16 Remark. Since φ is a norm, one readily checks that if a ∈ K, v ∈ X and δ φ K (a + v) = φ(v), then δ φ K (a + tv) = tφ(v) for every 0 ≤ t ≤ 1. 2.17 Remark. It has been observed in [DRKS20, 2.38(g)], using strict convexity of φ, that if a ∈ K, u ∈ ∂U φ (0, 1), 0 < t < ∞ and δ φ K (a+tu) = t, then ξ φ K (a+su) is a singleton and ξ φ K (a+su) = {a} for every 0 < s < t.
2.18 Definition (cf. [Roc70,p. 213]). Let f : X → R and x, v ∈ X. The one-sided directional derivative of f at x with respect to v is defined to be whenever the limit exists in R.

2.19
Remark. If f is a convex function and x is a point with f (x) ∈ R, then f (x; v) exists for every v ∈ X; cf. [ ). Suppose f : X → R is convex and x ∈ X is such that f (x) ∈ R. We say that ζ ∈ X is a subgradient of f at x if The set of all subgradients of f at x is denoted by ∇f (x).
2.21 Remark. Since the symbol "∂" is used in this paper for the topological boundary of a set and, on grounds of set theory, functions are sets it would introduce ambiguities if we used the standard notation "∂f " for the subgradient mapping of f ; hence, we decided to denote it "∇f ".
In the next definition we use the notion of a polynomial function which is formally defined in [Fed69, 1.10.4].
2.22 Definition. Let X, Y be normed vectorspaces and f be a function mapping a subset of X into Y . We say that f is pointwise differentiable of order k at x if there exist: an open set U ⊆ X such that x ∈ U ⊆ dmn f and a polynomial function P : X → Y of degree at most k such that f (x) = P (x) and lim y→x |f (y) − P (y)| |y − x| k = 0 .
Whenever this holds P is unique and the pointwise differential of order i of f at x, for i = 1, . . . , k, is defined by pt D i f (x) = D i P (x). As usual pt D 1 f (x) = pt Df (x).
2.23 Remark. We need to extend the concept of continuity and differentiability to multivalued maps.
2.25 Definition (cf. [Zaj83b, Definition 2]). Let X and Y be normed vectorspaces and T be a Y -multivalued map defined on X. We say that T is weakly continuous at a ∈ X if and only if T (a) = ∅ and for each ε > 0 there exists δ > 0 such that If, additionally, T (x) is a singleton, then we say that T is continuous at x.
2.26 Remark. We notice that if T (y) = ∅ for y ∈ B(x, δ) ∼{x} then T is continuous at x. On the other hand, we remark that studying the map ξ φ K we do not need to worry about such strange behaviour. Moreover, in 2.41(f) we prove that ξ φ K is weakly continuous on the whole of R n . Obviously, ξ φ K (x) is a singleton for all x ∈ X if and only if K is convex. 2.27 Remark. Note that weakly continuous multivalued functions may carry connected sets into disconnected sets. Consider, e.g., the function f : R → 2 R given by ; then, f is weakly continuous in the sense of 2.25. Another example is ξ φ K which is weakly continuous on the whole of R n regardless of the choice of the closed set K ⊆ R n ; in particular, when K is disconnected; cf. 2.41(f).

Definition (cf. [Zaj83b, Definition 3])
. Let X, Y be finite dimensional normed vectorspaces and T be a Y -multivalued map defined on X. We say that T is differentiable at a ∈ X if and only if T (a) is a singleton and there exists a linear map L : X → Y such that for any ε > 0 there exists δ > 0 satisfying The set of all such L is denoted by DT (a). In case DT (a) is a singleton, we say that T is strongly differentiable at a.
2.30 Remark. Let P and Q be two multivalued functions and x ∈ R n . If P is differentiable at x and Q is differentiable at P (x) then the multivalued function R given by 2.32 Remark. Definition 2.31 is well posed, since 2.17 gives that if ξ φ K (x) is not a singleton, then The following Lemma will be used in section 4.
2.33 Lemma. For every closed set K ⊆ R n the function ρ φ K is upper semicontinuous and satisfies we may, possibly choosing a subsequence, assume that lim i→∞ a i = a 0 and then a 0 ∈ ξ φ K (x 0 ) by continuity of both δ φ K and φ. Assume further that Noting that x = a + 1 t (y − a), ρ φ K (y) ≥ 1 t , and a ∈ ξ φ K (y), we can apply the inequality in (9), replacing x and t with y and 1 t respectively, to obtain the reverse inequality; hence, equality. Finally the assertion about the cut locus follows directly from the definition of ρ φ K .
again by Lemma 2.33. We conclude that x ∈ K σ and, by a direct computation, h t (x) = y. It follows that Since 0 < 1 t < σ t we apply the statement proved in the last paragraph with t and σ replaced by 1 t and σ t respectively to infer that h 1/t • h t = I Kσ and h 1/t [K σ/t ] = K σ . This proves that h t |K σ is an homeomorphism onto K σ/t . The next lemma provides an alternative description of the normal bundle N φ (K) defined in (4) and the reach function defined in (5).

Lemma. For every closed set
and Proof. Assume this is not true, so that for Let s ∈ R be such that The second part of the statement follows mechanically from the definitions.
2.36 Remark. Notice that in [DRKS20, Remark 5.6] we erroneously claim that r φ K is lower semicontinuous which is obviousy wrong but, fortunatelly, does not affect other results of [DRKS20] since we only need the fact that r φ K is a Borel function there. 2.37 Remark. The function r φ K can fail to be continuous even if K is a compact convex C 1,1 hypersurface. In fact in [San21] we show that there exists a compact and convex C 1,1 -hypersurface K such that Clos(Σ(K)) has non empty interior. Noting that N (K) is the classical unit normal bundle of K and consequently it is compact, we infer that if r K was continuous then Cut(K) would be compact; consequently Clos(Σ(K)) = Cut(K) and L n (Cut(K)) > 0 which is incompatible with Remark 4.4.

Auxiliary results
The following lemma shows that if A ⊆ R n is a set of points at which a multivalued function f satisfies a Lipschitz condition, a is a density points of A, and f |A is differentiable at a, then f is differentiable at a. It is a variant of a classical result stating that a Lipschitz function that is approximately differentiable at a point is classically differentiable at that point; cf. [Fed69, 3.1.5].
2.38 Lemma. Assume Then f is strongly differentiable at a.
Proof. Since a is a density point of A we see that f |A is strongly differentiable at a and Df (a) = {L} for some L ∈ Hom(R n , R n ); cf. 2.29. Let ε > 0. Choose 0 < δ < ε such that Clearly B(c, |c − b|) ⊆ B(a, 2r) and L n B(c, |c − b|) = α(n)|c − b| n ; hence, The next lemma is a classical result in convex analysis.
2.39 Lemma. If U ⊆ R n is an open convex set, f : U → R is a convex function and x ∈ U , then the following three statements are equivalent.
(a) f is pointwise differentiable of order 2 at x.
(c) There is at least one function g : U → R n such that g(y) ∈ ∇f (y) for every y ∈ U and g is differentiable at x.

If (a), (b), and (c) hold, then
Proof. Clearly ∇f (y) = ∅ for all y ∈ U because f is convex and 2. 24 2.40 Definition. Suppose U ⊆ R n is open. We say that a function g : U → R is semiconcave if and only if there exists κ ≥ 0 such that the function g(y) − (κ/2)|y| 2 is concave.
The following lemma collects few facts on the continuity, differentiability, and convexity properties of δ φ K and ξ φ K for an arbitrary closed set K. 2.41 Lemma. Let K ⊆ R n be a closed set. Then the following statements hold.
It follows that |a i − a| ≥ ε for every i ≥ 1, which is in contradiction with a i → a.
2.42 Remark. Continuity properties of ξ φ K |U will be studied more carefully in 3.2 in case φ is strictly convex and in 3.9 in case φ is uniformly convex. Proof. We prove that Df (α) exists and equals zero. If lim sup T χ→α |f (χ) − f (α)| · |χ − α| −1 > 0, then we could find a sequence χ j ∈ T such that χ j → α, (χ j − α) · |χ j − α| −1 → w ∈ T , and (f ( The following Lemma follows rather directly from classcial implicit function theorems for Lipschitz and semiconcave functions. In the next lemma, given x ∈ R n , , δ > 0, and a linear space T ⊆ R n , we make use of cylinders aligned to T defined the following way Moreover, we recall from Lemma 2.41 that . Then T = Tan(S φ (K, r), x) and there are , δ > 0 and a semiconcave function f : Moreover, if δ φ K is pointwise differentiable of order 2 at x then f is pointwise differentiable of order 2 at T x and Proof. We notice that δ φ K is locally semiconcave on R n ∼ K by Lemma 2.41(b). Since δ φ K is differentiable at x and grad δ φ K (x) = 0, noting Remark 2.24 and [Fu85, Remark 1.4], we see that we can apply [Fu85, Theorem 3.3] to find , δ > 0 and a semiconcave function f : T → R such that (10) and (11) hold 1 . Since δ φ K is differentiable at x, then Tan(S φ (K, r), x) ⊆ T . Therefore, the first part of the conclusion follows from Lemma 2.43.
1 At a first sight we can only deduce from [Fu85, Theorem 3.3] that there exist , δ > 0, an hyperplane S ⊆ R n and a semiconcave function f : S → R such that (10) and (11) with S replaced by T . However, closer inspection of the proof of [Fu85, Theorem 3.3] reveals that we can choose S = T , as the existence of a lipschitzian function f : T → R which satisfies (10) for some , δ > 0 directly follows from Clarke implicit function theorem.

Lipschitz estimates
In this section we consider an abstract Minkowski space (X, φ) of dimension n and we are defining a Euclidean structure on X to fit our problem. For this reason we choose to denote the space with "X" rather than "R n " since the latter refers to a space with a predefined Euclidean structure which is of no use to us. The operator norm of o bilinear map Λ : X × X → X with respect to φ is defined as in [Fed69,1.10.5], i.e., Once the Euclidean structure on X is defined we shall use the symbol Λ to denote the operator norm of Λ with respect to that Euclidean structure. (cf. [Fed59, 4.1]). Let K ⊆ X be closed. We define the set of points with unique nearest point

Definition
We start by showing that ξ φ K is uniformly continuous on certain sets. Later, in 3.9 and 3.10, we bootstrap this regularity to Lipschitz continuity. Uniform continuity is obtained for strictly convex norms φ, while Lipschitz continuity requires uniform convexity and C 2 regularity of φ.

Remark.
Assume that X is a finite dimensional vectorspace equipped with a strictly convex and continuously differentiable (away from the origin) norm φ : X → R. We define π : S → Hom(X, X) by π = Dξ|S .
3.7 Remark. Consider the situation as in 3.5 and assume φ is of class C 2 away from the origin. Let ε ∈ (0, 1) and η ∈ S. Set R = R 3.6 (ε), T = Tan(S, η), and M = S ∩ B φ (η, R). Since π(η)|M is injective and M is compact we see that π(η)|M is a homeomorphism between M and A = π(η)[M ] ⊆ T . Set Since φ is of class C 2 we see that M is a manifold of class C 2 and H : C → M is of class C 2 , H(ζ) = ξ(ζ) and DH(ζ)u = Dξ(ζ)u for ζ ∈ S ∩ C and u ∈ Tan(S, ζ) .
Differentiating the equation however, if u, v ∈ T = im π(η), then Dπ(η)uv ∈ ker π(η) = span{η} by (13) and for all x ∈ S ∩ C we also have DH(x)η = 0; hence Since T is tangent at η ∈ S to the level-set S of φ we have Dφ(η)u = 0 whenever u ∈ T ; thus, differentiating (12) twice and recalling that φ(η) = 1 and ξ(η) = η we obtain 3.8 Remark. In 3.9 we prove that ξ φ K is Lipschitz continuous on each of the sets K λ,s,t = {x : ρ φ K (x) ≥ λ , s ≤ δ φ K (x) ≤ t} defined for 0 < s < t < ∞ and 1 < λ < ∞. Since the proof is a bit technical we briefly describe the main idea. For x ∈ K λ,s,t and y ∈ R n ∼ K with φ(x − a) ≤ ε we set a = ξ φ K (x) and choose any b ∈ ξ φ K (y). First we find a point c for which T = Tan(∂B φ (x, δ φ K (x)), a) = Tan(∂B φ (y, δ φ K (y)), c). For this point we have φ(a − c) ≤ 2φ(x − y); see (17). Then we choose e ∈ ∂B φ (y, δ φ K (y)) and d ∈ ∂B φ (a + λ(x − a), λδ φ K (x)) which have the same orthogonal (with respect to the Euclidean structure induced by D 2 φ(a−x)) projections onto T as a and b respectively; see Figure 1. We represent ∂B φ (a + λ(x − a), λδ φ K (x)) and ∂B φ (y, δ φ K (y)) locally around a and c as graphs over T of functions g w and g u of class C 2 using 3.6. Employing 3.2 we can find ε > 0 which guarantees that d, e, and b fit on the graphs of g w , g y , and g y respectively. Let q be the signed distance from T such that q(x − a) > 0. The crucial point of the proof is in the estimates (21) and (22), where we use the second order Taylor formulas for g w and g y to compare (both ways) the heights q(d − a), q(e − c), and q(b − c) with λ −1 |T (d − a)| 2 , |T (a − c)| 2 , and |T (b − c)| 2 respectively up to errors expressed in terms of the modulus of continuity of D 2 H, where H comes from 3.7. Analysing the situation presented on Figure 1 we obtain an estimate of the form which, using the comparison mentioned before, is translated into where ∆ 1 and ∆ 2 can be made arbitrarily close to 1 by adjusting ε depending on the modulus of continuity of D 2 H. This leads to the estimate (24) of the form where, again, ∆ 4 is close to 1 given ε is small enough; hence, the last term may be absorbed on the left-hand side. Since |T (b − a)| ≈ |b − a| and |T (c − a)| ≈ |x − y| we get the conclusion.
By our assumption on D 2 φ(η) the map B defines a scalar product on X. In the sequel of this proof we shall assume the Euclidean structure on X comes from B. In particular, we shall use the notations Let ω λ be the map obtained from 3.2. Set where the operator norm of the bilinear map D 2 H(ζ) − D 2 H(χ) : X × X → X is taken with respect to the Euclidean structure on X defined by (14). Choose ε ∈ R so that Figure 1: We introduce a Euclidean structure on X so that x − a is orthogonal to T .

Twice differentiability points
In this section we prove Theorem 1.5. Recall that r φ K was defined by (5), pointwise differentiability in 2.22, φ-cut locus Cut φ (K) by (6), and singular sets Σ φ (K) and Σ φ 2 (K) in (1) and (3). 4.1 Remark. It is well known, and follows from 2.17, 2.41(c) and [Fre97,Theorem 3B], that 4.2 Remark. Consider the parabola K = {(x, x 2 ) : x ∈ R} with centre of curvature at the point a = (0, 1 2 ) ∈ R 2 . Then a ∈ Cut(K) ∩ Unp(K). We look at the behaviour of ξ K on the line {(x, 1 2 ) : x ∈ R}. Whenever 0 < x < 8 −1/2 , setting b = (2x, 1 2 ), we have ξ K (b) = ( √ x, x); hence, ξ K is not differentiable at a and δ K is not pointwise differentiable of order 2 at a. Note also that ξ K is not even Lipschitz continuous in any neighbourhood of a. On the other hand 2.41(c) yields differentiability of δ K at a (which can also be checked by direct computation). We conclude a ∈ Σ 2 (K) ∼ Σ(K). In 4.3 we prove that this is a generic situation for points in Cut(K) ∩ Unp(K).
4.3 Lemma. Assume K ⊆ R n is closed, x ∈ R n ∼ K, and δ φ K is pointwise differentiable of order 2 at x.
We use 2.44 to find r 1 > 0 and a continuous function f : T → T ⊥ which is pointwise twice differentiable at T x with Df (T x) = 0 such that, defining M = {χ + f (χ) : χ ∈ T } and U = U φ (x, r 1 ), it holds U ∩ S φ (K, r) = U ∩ M . Decreasing r 1 > 0 if necessary, we infer from the pointwise twice differentiability of f in T x that there exists a polynomial function P : T → T ⊥ of degree at most 2 such that Decreasing r 1 > 0 even more, we can assume also that U ∼ S φ (K, r) is the union of two connected and disjointed open sets U − and U + such that (27) that there exists s > 0 such that U φ (x + sν, s) ⊆ U + (notice s < r 1 ) and Choose 0 < < r1 4 . The continuity of ξ φ K and δ φ K at x implies that there exists 0 < δ < such that φ(b − a) < and φ(b − y) < r + for every b ∈ ξ φ K (y) and for every y ∈ U φ (x, 2δ). Define y = x + δν, choose b ∈ ξ φ K (y) and let τ = sup{t : Therefore, y + τ (b − y) ∈ U ∩ B φ (a, r) ⊆ Clos U − ∩ U . Since y ∈ U + we infer there exists 0 < t ≤ τ such that y + t(b − y) ∈ S φ (K, r). Defining z = y + t(b − y) and noting that φ(z − b) = φ(x − a) and φ(y − z) ≥ φ(y − x) by (28), we infer whence we conclude that a ∈ ξ φ K (y) and consequently ρ φ K (x) > 1.
In a Riemannian setting a conclusion analogous to Lemma 4.3 is contained in [Alb15]. A proof of L n (Cut φ (K)) = 0 along different lines can be found in the proof of [DRKS20, Theorem 5.9, Claim 1], see also [DRKS20,Remark 5.10].
In the next result the classical notion of approximate lower limit of a function plays a central role. Let us first recall this definition.
The approximate lower limit of an arbitrary function always defines a Borel function. This fact can be proved using an argument similar to those of [San19, Lemma 5.1].

4.7
Lemma. Suppose f : R n → R is an arbitrary function and let f : R n → R be defined as Then f is a Borel function.
Proof. For every t ∈ R we define F t = {x : f (x) < t} and we set for t ∈ R, i ∈ Z + and r > 0. Then we prove that the set W t,i,r is a closed subset of R n for every t ∈ R, i ∈ Z + and r > 0. Choose a sequence y k ∈ W t,i,r that converges to y ∈ R n . Noting that and y ∈ W t,i,r . Fix now σ ∈ R, an increasing sequence t j converging to σ, and a countable dense subset D of R. Noting that we conclude that {x : f (x) ≥ σ} is a Borel subset of R n ; hence, f is a Borel function.
We consider now the approximate lower envelope of ρ φ K (see Definition 2.31).
4.8 Definition. For a closed set K ⊆ R n we define the function ρ φ K : R n → R as 4.9 Remark. Clearly 1 ≤ ρ φ K (x) ≤ lim sup y→x ρ φ K (y) ≤ ρ φ K (x) for x ∈ R n by Lemma 2.33. 4.10 Remark. Let (a, η) ∈ N φ (K). If there exists 0 < r < r φ K (a, η) and σ > 1 with ρ φ K (a+rη) ≥ σ, then it follows from Remarks 4.6 and 4.9 and Lemma 2.35 that Proof. The function ρ φ K is a Borel function by Lemma 4.7. Let h t be defined as in Lemma 2.34 for all t ∈ R. Suppose x ∈ R n ∼ K, σ = ρ φ K (x) > 1 and 0 < t < σ. We choose 0 < < and we notice that and, with the help of Lemma 2.34, Then we infer from Corollary 3.10 and Lemma 2.34 that h t |K σ ∩ B φ (x, ) is a bi-Lipschitz homeomorphism. Since Θ n (L n K σ , x) = 1, we employ [Buc92, Theorem 1] to conclude that Noting that ρ φ K (h t (x)) ≥ ρ φ K (x)/t > sup{1, 1/t}, we can apply the inequality in (29), with x and t replaced by h t (x) and 1 t respectively, to obtain the desired conclusion. 4.12 Lemma. Suppose Then the following statements hold.
(d) Dh t (x) is an isomorphism of R n for every 0 < t < λ.
(e) If σ > 1 the R n -multivalued map h 1/t is strongly differentiable at h t (x) for every 0 < t < σ.
Then T is continuous at h t (x) and S = T −1 by Lemma 4.11 (see Definition 2.13). Moreover, since Θ n (L n K σ , x) = 1, it follows from Remark 2.29 that S is strongly differentiable at x with DS(x) = Dh t (x) and it follows from Lemma 4.11 that Θ n (L n K σ/t , h t (x)) = 1.
We apply [Zaj83b, Lemma 2] to conclude that T is differentiable at h t (x) and, noting the Lipschitz property of h 1/t over K σ/t that can be deduced from Corollary 3.10, we can use Lemma 2.38 to conclude that h 1/t is strongly differentiable at h t (x). Therefore ξ φ K is strongly differentiable at h t (x) and δ φ K is pointwise differentiable of order 2 at h t (x) by Lemma 2.41(e). 4.13 Remark. The proof of (c) shows the following fact. Suppose ξ : R n → R n is an arbitrary function such that ξ(y) ∈ ξ φ K (y) for y ∈ R n and η(y) = grad φ(y − ξ(y)) | grad φ(y − ξ(y))| for y ∈ R n ∼ K.
4.14 Lemma. Suppose K ⊆ R n is closed and (a, η) ∈ N φ (K). Then the following statements are equivalent.
We turn to the proof of (c). Set B = (R n ∼ K) ∩ {x : ρ φ K (x) < ρ φ K (x)}. Since ρ φ K is a Borel function by 2.33 and ρ φ K is L n measurable by Lemma 4.11 we see that B is L n measurable and it follows from [Fed69,2.9.13] that L n (B) = 0; hence, the coarea formula [Fed69, 3.2.11] yields a set J ⊆ R + such that L 1 (R + ∼ J) = 0 and H n−1 (B ∩ S φ (K, t)) = 0 for t ∈ J .