Anisotropic tubular neighborhoods of sets

Let E⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset {{\mathbb {R}}}^N$$\end{document} be a compact set and C⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subset {{\mathbb {R}}}^N$$\end{document} be a convex body with 0∈intC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in \mathrm{int}\,C$$\end{document}. We prove that the topological boundary of the anisotropic enlargement E+rC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E+rC$$\end{document} is contained in a finite union of Lipschitz surfaces. We also investigate the regularity of the volume function VE(r):=|E+rC|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_E(r):=|E+rC|$$\end{document} proving a formula for the right and the left derivatives at any r>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>0$$\end{document} which implies that VE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_E$$\end{document} is of class C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} up to a countable set completely characterized. Moreover, some properties on the second derivative of VE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_E$$\end{document} are proved.


Introduction
The study of the tubular neighborhood E r := {x ∈ R N : dist(x, E) ≤ r } of a convex set E in R N plays a crucial role in convex geometry. Of course, is not without interest to investigate the tubular neighborhood also for non convex sets, and it turns out that the boundary of E r becomes more regular than the boundary of E, which could be very irregular: more precisely, in 1985 Fu [11] proves that ∂ E r is a Lipschitz manifold whenever E is compact in R N and r > r 0 for some r 0 > 0. The approach of Fu is essentially based on the fact that the sublevels of regular values of a proper and semiconcave function are sets of positive reach: this argument can be applied since the distance function is semiconcave far from E. The semiconcavity of the distance is strongly related with the smoothness of the ball in R N : notice indeed that E r can also be written as E r = E + r B, where B is the unit closed ball centered in the origin. In this paper first of all we investigate the extension of such results to the anisotropic case, that is in the case E r = E + rC where C is a prescribed convex body, i.e. a compact convex set in R N with 0 ∈ int C. In this case the appropriate anisotropic distance to E, which we denote by d E , could not be semiconcave outside E, since we are not assuming any kind of regularity of the boundary of C, unless locally Lipschitz coming from convexity: notice that we are really interested in enlarging E with a convex body C, since this case recovers also the crystalline anisotropy where C is convex but not necessarily strictly convex nor smooth. We will prove (see Thm. 3.1) that for any r > 0 the boundary of E r is contained in a finite union of Lipschitz surfaces when E is bounded and C is Lipschitz with 0 ∈ int C. Of course since C is not sufficiently smooth we cannot use the Fu's approach, but the key idea of our proof is very easy: we first prove that enlarging C by a very small set, like εK with ε > 0 small and K ⊂ B, we still obtain a Lipschitz domain, and then we use the same idea of Rataj and Winter [17] covering ∂ E r by a finite union of sets with small diameter. The rectifiability of ∂ E r is an independent interesting result, but actually we need to prove the regularity of ∂ E r in order to study the regularity of the volume function V E (r ) := |E r | (see [21,22] for the isotropic case). We therefore characterize the set, at most countable, where V E is not differentiable (see Thm. 5.2) and we find explicit formulae for left and right derivatives of V E . Moreover, V E is of class C 1 whenever it is differentiable (see Thm. 5.3). We mention that such a result finds application also in different fields, for example, in Stochastic Geometry, where V E is strictly related to the notion of covariogram and of contact distribution function associated to a random closed set (e.g. , see [23,Sec. 4] for the isotropic case, and the recent paper [15] where dilation by finite sets is considered). Finally, an easy characterization of V E is proved (see Thm. 5.4). Our result is a generalization of the isotropic case [14] and our proof is partially based on the so called anisotropic outer Minkowski content (see [6] and [16] for details). We also need to base our argument on the existence of a so called Cahn-Hoffmann vector field for C with divergence measure bounded from above (see Prop. 4.1), and this is, in our opinion, an interesting result independent on the rest, since we are not assuming the strict convexity of C, so that such an existence result holds true also in the crystalline case.

Notation
For any A subset of R N we will denote by |A| the Lebesgue measure of A while H k (A) stands for the k-dimensional Hausdorff measure of A, where k ∈ {0, . . . , N }; of course H N is the Lebesgue measure. For any x ∈ R N the euclidean norm of x will be denoted by |x| while x · y stands for the euclidean scalar product in R N between x and y. For any r > 0 and x ∈ R N the closed ball centered in x with radius r will be denoted by B r (x); we let B r := B r (0) and S N −1 := ∂ B 1 . We finally denote by ω k the volume of the k-dimensional unit ball in R k .

Convex analysis
Here we recall some basic notions of convex analysis; for all details we refer to [18]. In this paragraph C will be a convex body in R N , that is a compact convex subset of R N with 0 ∈ int C. We denote by h C : R N → R the support function of C, that is h C (v) := max x∈C x · v. We will use also the polar of h C , denoted by h • C and defined by h x · v for each v ∈ R N ; it turns out that both h C and h • C are convex and positively 1-homogeneous. We will need also to consider convex sets for which the support function and its polar are more regular. Let C be of class C 2 . We say that C is elliptic if the curvature of ∂C is bounded from below by some positive constant. It turns out that if C is C 2 and elliptic then both h C and h • C are in C 2 (R N \{0}). A very useful notion related with convexity is given by semiconcavity. Let A be a subset of R N and let f : A → R. We say that f is concave if the inequality is said to be semiconcave if there exists α > 0 such that for any x, y ∈ A and for any λ Notice that if f is semiconcave and smooth enough, for instance of class C 2 , then D 2 f ≤ α I , where I is the identity matrix and the inequality holds in the sense of matrices. A useful class of semiconcave functions can be constructed; we have the following well known proposition, see for instance [9].
satisfies (2.1) uniformly with respect to s.

Geometric measure theory
In this paragraph we recall some notions of Geometric Measure Theory we will need; for all details we refer the reader to [2], [10] and [20]. Let N ≥ 1 be integer and let k ∈ N with k ≤ N . Let S ⊂ R N . We say that S is k-rectifiable if there exist a bounded set B ⊂ R k and a Lipschitz function f : B → R N such that S = f (B); equivalently, by the Kirszbraun's extension Theorem, we can say that S is k-rectifiable if S is contained in a finite union of Lipschitz surfaces in R N . We say that S ⊂ R n is countably H k -rectifiable if there exist countably many Lipschitz functions f h : It turns out that if S is countably H k -rectifiable then for H k -almost any point x 0 ∈ S it is well defined the approximate tangent space Tan k (S, x 0 ), that is In particular, if k = N − 1 then Tan N −1 (S, x 0 ) ⊥ is generated by some unit vector denoted by ν S . Let now E ⊂ R N be a measurable set and ⊂ R N be an open domain; we denote by χ E the characteristic function of E. We say that E has finite perimeter in if χ E ∈ BV ( ); the perimeter of E in is defined by P(E; ) := |Dχ E |( ), where |Dχ E | denotes the total variation of Dχ E ; we also let P(E) := P(E; R N ). For sufficiently smooth boundaries the perimeter coincides with the (N − 1)-dimensional Hausdorff measure of the topological boundary. The upper and lower N -dimensional densities of E at x are respectively defined by * . It turns out that if E has finite perimeter in , then H N −1 (∂ * E\E 1/2 ) = 0, and P(E; ) = H N −1 (∂ * E ∩ ). Moreover, one can define a subset of E 1/2 as the set of points x where there exists a unit vector ν E (x) such that and which is referred to as the outer normal to E at x. The set where ν E (x) exists is called the reduced boundary and is denoted by Let us collect some elementary properties of sets with countably H N −1 -rectifiable boundary and with finite perimeter in ; for any E ⊆ R N we let E c := R N \E. Assume that E has finite perimeter in and ∂ E is countably H N −1 -rectifiable. Then the following relations hold true: We finally recall an anisotropic version of the coarea formula which we will need; for details see [13,Thm.3]. Let u ∈ BV ( ) and let α : R N → (0, +∞) be a convex and positively one-homogeneous function with c −1 |v| ≤ α(v) ≤ c|v| for any v ∈ R N and for some constant c > 0. Then the following formula holds true:

Anisotropic outer Minkowski content
We now briefly recall the notion of outer Minkowski content; for details see [1] and [22] (isotropic case), [6] and [16] (anisotropic case). Let C ⊂ R N be a convex body. For each closed set E ⊂ R N we also define the anisotropic outer Minkowski content of E as whenever such a limit exists.
The following existence and characterization result for SM C holds true.
Theorem 2.2 [16,Thm. 4.4] Let E ⊂ R N be a closed set such that: (b) there exist γ > 0 and a probability measure η in R N absolutely continuous with respect to H N −1 such that η(B r (x)) ≥ γ r N −1 for all x ∈ ∂ E and for all r ∈ (0, 1).
Notice that any compact set in R N whose boundary is contained in a finite union of Lipschitz surfaces satisfies property (b): see for instance [1,Rem. 1]. We also observe that even if ν E is not well defined on ∂ E ∩ E 0 , the expression φ C (ν E ) turns out to be well defined.

Anisotropic tubular neighborhoods
In this paragraph we will introduce all the objects we want to investigate. Let N ≥ 1 be integer. Let E ⊂ R N be compact and C ⊂ R N be a compact Lipschitz set with 0 ∈ int C. For any r > 0 denote E r := E + rC. Moreover, let It is convenient to introduce the anisotropic distance from E, that is Notice that E r = {d E ≤ r } and E r = {d E < r }. It turns out (for details see [6]) that d E is Lipschitz continuous and, if C is a convex body, Note that for C = B 1 , it is also named volume function of E (see also [21,22]). It is easy to see that V E is continuous.

Regularity of the boundaries
In this section we prove that ∂ E r and ∂ E r are sufficiently smooth, in the sense of geometric measure theory. Proof We divide the proof in two steps.
Step 1: Let K ⊂ R N be a bounded set. We claim that for ε positive and sufficiently small the set C + εK is a Lipschitz set.
Without loss of generality we can assume K ⊂ B 1 . For any ξ ∈ R N , ξ = 0, we let where π ξ denotes the orthogonal projection on ξ ⊥ . Since C is Lipschitz and compact we can write its boundary locally as a graph of a Lipschitz function in a uniform way: precisely, we can find r > 0 such that B r ⊂ C and such that for any z ∈ ∂C there exists a Lipschitz function f z : Let ε < r /2 and fix x 0 ∈ ∂(C + εK ). There exists k 0 ∈ K such that x 0 ∈ ∂C + εk 0 , thus For any x ∈ B r /2 ∩ z ⊥ 0 and any k ∈ K let: For ξ ∈ ∂(C + εK ) ∩ S z 0 r /2 , writing ξ = η + εk, η ∈ ∂C, k ∈ K , we observe that for y = π z 0 (η) one has η = y + f z 0 (y)ẑ 0 with |y| ≤ r . Thus, one finds that ξ decomposes as and it follows that: , and then η n = ξ − εk n ∈ ∂C). On the other hand if one lets now ξ n : We notice eventually that g is Lipschitz continuous with the same Lipschitz constant L of f z 0 , which achieves the proof that ∂(C + εK ) is locally a Lipschitz graph: indeed for any Step 2: Now it is relatively easy to conclude the proof for ∂ E r ; the rectifiablity of ∂ E r follows since ∂ E r ⊆ ∂ E r . The idea is to use the same argument as in the proof of [17,Prop. 2.3]. If r > 0 by step 1 we can say that for any x ∈ R N the set rC + (B r (x) ∩ E) has Lipschitz boundary for r < r sufficiently small (apply step 1 to rC instead of C). We cover now E, which has compact closure, with balls B r ( that is ∂ E r is contained in a finite union of Lipschitz surfaces, and this yields the conclusion.

Construction of a Cahn-Hoffmann vector field for C
First of all, we recall some basic results of the theory of viscosity solutions; for details we refer to [8]. Let Sym N (R) be the set of all symmetric N × N matrices with real entries, let be a subset of R N and let F : × R × R n × Sym N (R) → R be a continuous function such that the following monotonicity condition holds: whenever r ≤ s and Y ≤ X in the sense of matrices. Let u : → R be upper semicontinuous. We say that u is a viscosity subsolution of the equation F(x, u, Du, D 2 u) = 0 on if for any φ ∈ C 2 ( ) and for anyx ∈ local maximum point of u − φ it holds Let now u : → R be lower semicontinuous. We say that u is a viscosity supersolution of the equation F(x, u, Du, D 2 u) = 0 on if for any φ ∈ C 2 ( ) and for anyx ∈ local minimum point of u − φ it holds If u is both a viscosity subsolution and supersolution then u is called viscosity solution of F(x, u, Du, D 2 u) = 0 on .
We are ready to start the construction of a Cahn-Hoffmann vector field for C in the smooth case.

Proposition 4.1 Assume that C is a convex body of class C 2 and elliptic. Let n
and div n is a Radon measure on in the distributional sense out of E r .
Proof First of all we point out that the assumptions on C guarantee that both h C and h • C are in C 2 (R N \{0}). From the standard fact that h 2 C /2 and (h • C ) 2 /2 are Legendre-Fenchel convex conjugates, so that their gradients h C ∇h C and h • C ∇h • C are inverse mappings, we deduce that for any z ∈ R N \{0} For the sake of simplicity we will denote d := d E .
Step 1. The proof of (4.1) is easy: indeed, if we fix from which we immediately get (4. Step 2. We prove (4.2). First of all, it turns out that d is a viscosity supersolution of in R N \E r . This is a variant of a classical result, see [3]. The proof is quite straightforward. Indeed, if φ is a smooth function which touches the graph of d from below at a point ) then by definition of d, φ also touches the graph of x → h • C (x −ȳ) from below atx, whereȳ ∈ E is a point of minimal distance tox. Being both functions smooth atx, it follows that ∇φ( . Combining (4.3) with the Euler's identity, for any z ∈ R N \{0} we obtain, also by direct computation, .
We find that not only d is a viscosity supersolution of (4.4) out of E r , but the more precise inequality holds. Since h • C ∈ C 2 (R N \{0}) by Proposition 2.1 we can say that d is (locally) semiconcave out of E r , and in particular D 2 d ≤ c in both the viscosity and distributional sense. It is not obvious however to deduce from these facts that out of E r in the sense of distributions, as the left-hand side is the product of a L ∞ , yet discontinuous function, and a Radon measure.
We pick now R > r , λ > 0, and we introduce u λ a solution of the problem Notice that we can easily apply on the functional in (4.5) direct method of the Calculus of Variations: we have lower semicontinuity in the strong convergence of L 1 essentially by Reshetnyak's lower semicontinuity and we have strong L 1 -compactness of sequences bounded in energy since h C (v) ≥ c|v| for some c > 0. Moreover, observe that by truncation arguments we clearly have r ≤ u λ ≤ R in E R \E r . Standard density estimates for the level sets of u λ show also that u λ is a.e. equal to a lower and a upper-semicontinuous function.
We assume that u λ is upper-semicontinuous, and is a.e. equal to its lower-semicontinuous envelope. We check then that u λ is a strict viscosity subsolution of (4.4) in {u λ > d}, in the following sense: if φ ≥ u λ , φ smooth, φ(x) = u λ (x), then if ∇φ(x) = 0 one has The proof is easy and quite standard. Possibly replacing φ with φ +η|·−x| 2 , η small, we may assume thatx is the only contact point. Then, one checks that {φ − δ < u λ } has nonempty interior and goes to {x} in the Hausdorff distance as δ → 0. We denote H λ = (N − 1)/r + λ.
For δ > 0 small we have

Moreover, since for any open set
satisfies the generalized coarea formula (2.3) and it is convex, we get submodularity (see [5,Prop. 3

.2]), which reads as
Therefore, we obtain that (letting A a small open set containing {φ − δ < u λ }, for δ small) If ∇φ(x) = 0 then one may assume that ∇φ = 0 in A, so that it follows We deduce that div ∇h C (∇φ)(x) ≥ H λ , as claimed, otherwise one reaches a contradiction for small δ. Now, we can deduce that u λ ≤ d (so that in particular u λ = d), using a standard comparison result for viscosity sub and supersolution (with one possibly discontinuous). We sketch the argument, see [4] and [8] for details. Let m := max{u λ − d} and assume by contradiction that m > 0. For δ > 0 small, we consider is a sup-convolution. In particular, if x ∈ {u λ δ > d + m/2}, a pointȳ which reaches the maximum in (4.6) is such that u λ (ȳ) > d(ȳ) as soon as δ < m/L 2 (L denoting the Lipschitz constant of d), and in this case u λ δ is still a strict subsolution of (4.4) in {u λ δ > d + m/2}: as a test function in the definition of strict subsolution of (4.4) applied to u λ . Now, since u λ δ is (near x δ ) semiconvex while d is semiconcave, we can invoke Jensen's Lemma (see [8] for details), and find that there are points x n → x δ which are local maximum points of with p n → 0, α n → 0, u λ δ (x n ) > d(x n ) + m/2; notice that we have to add the term α n |x−x δ | 2 2 since, in order to apply Jensen's Lemma, we need x δ be a strict local maximum of the function we perturb with the linear term p n · x. By Aleksandrov's Theorem (see again [8] for details) we can also assume that u λ δ and d are both twice differentiable at x n . In particular, for n large where o(1) → 0 as n → ∞. Since λ > 0 this yields a contradiction. Hence u λ = d for any λ > 0, and it follows that d is the only minimizer of (4.5) for any λ > 0, and in the limit is also a minimizer for λ = 0. Finally, we have shown that the functional in (4.5) is minimized by d, including for λ = 0. But then, the Euler-Lagrange equation for the problem is easily derived: using perturbations d +δφ with δ > 0 small, φ smooth, nonnegative, with compact support in E R \E r , we readily find that is precisely (4.2) in the distributional sense.
We are ready to prove essentially the same result stated in Proposition 4.1 for a general convex body C.

Theorem 4.2 Let C be a convex body. There exists n
and div n is a Radon measure on R N \E with div n ≤ N − 1 r (4.8) in the distributional sense out of E r .
Proof We use again the notation d = d E . We prove (4.7) and (4.8) approximating C by smooth, elliptic, uniformly bounded and convex sets C σ , with C σ ⊇ C, and using Proposition 4.1. Let E σ r := E + rC σ and denote by d σ the anisotropic distance from C σ . Then n σ := ∇h C σ (∇d σ ) ∈ C σ is well defined a.e.,and (4.2) reads div n σ ≤ N − 1 r (4.9) out of E σ r . As σ → 0 + we can assume, up to a subsequence, since ||n σ || ∞ remains bounded by (4.1), that n σ * n in L ∞ (R N ; R N ) and we have for any nonnegative C 1 function φ with compact support in R N \E r , for σ small enough (using the Hausdorff convergence of E σ r to E r ), as σ → 0 + , showing that in R N \E r , div n is a measure bounded from above by (N − 1)/r , so that we get (4.8). On the other hand, if η : R + → R is any smooth nonincreasing function with η(t) = 1 for t ≤ r , η(t) = 0 for t large, one has (since n σ = ∇h C σ (∇d σ ) ∈ ∂h C σ (−η (d σ )∇d σ ), using that ∇h C σ is zero-homogeneous and always contained in ∂h C σ (0)): Since h C σ ≥ h C , we easily see that, from η • d σ → η • d in any L p and using standard lower semicontinuity results for integral functionals, On the other hand (using (4.9)), which together with (4.10) yields Since n ∈ C a.e. we obtain (4.7) and this ends the proof.  [12] provided a construction of an n satisfying (4.7) with minimal |div n| 2 . Also, an alternative way to build a Cahn-Hoffmann field satisfying (4.7) can be deduced from the construction in Chambolle, Morini and Ponsiglione [7], in addition this should also provide a field with minimal curvature.

Regularity of the volume function
In this section we investigate the regularity of the volume function V E . Our result extends [14,Eq. (2.20)], where an expression for V E has been given whenever C is strictly convex.
In what follows n is given as in Theorem 4.2. Let Remark 5. 1 We will prove (see (5.7)) that for any r > 0, In what follow we denote by V E (r + ) and V E (r − ) respectively the right and the left derivative of V E .

Theorem 5.2 For any r > 0 we have
and

In particular, V E is differentiable at r if and only if r / ∈ J .
Proof Notice that from the fact that ∂C is locally Lipschitz and compact we easily deduce that that is formula (5.1). It remains to compute the left derivative of V E . We divide the rest of the proof in some steps.
Step 1. Let C * := −C, that is the symmetrical of C with respect to the origin; notice that We also introduce the corresponding anisotropic distance to E r c : where we have denoted E r c := (E r ) c . Let s ∈ (0, r ). Notice that By definition there exist ε > 0 and z ε ∈ E r c such that Then, for any y ∈ E we obtain, by the subadditivity of h Taking into account Lemma 3.1 we can say that |{d * = s}| = 0 and |E c r | = |E r c |, hence Passing to the limit as s → 0 + we deduce that Using Theorem 2.2 we get Notice now that if r / ∈ J then H N −1 (∂ E r ∩ E 1 r ) = 0. We obtain that for any r ∈ (0, +∞)\J Step 2. We prove now that for any r > 0 lim sup For any s ∈ (0, r ) we have, using the coarea formula and (5.15), Therefore, by (5.9) we obtain lim sup which is (5.6).

Proposition 5.3 For any r
In particular, V E is C 1 in (0, +∞)\J .
Proof Let us prove (5.8). The easy part is the estimate from below: since Dχ E s * Dχ E r , as measures as s → r + , applying Reshetnyak's lower semicontinuity we have Now we divide the rest of the proof in some steps.
Step 1. We claim that for each continuous function ψ : (5.10) For simplicity of notation we set First of all, combining (2.5) with the coarea formula, for any positive integer k we obtain For a general continuous function ψ it is sufficient to apply the previous argument to ψ + and ψ − .
Step 2: Consider η : R → R a smooth nondecreasing function with η ≡ 1 on R − and η(t) = 0 for t ≥ 1. Then, letting, for k ≥ 0, ψ k (x) := η(k(d E (x) − r )) and ψ ε k (x) := η(k(d E (x) − r − ε)), one has, using (4.8), as k → +∞. On the other hand, using the definition of n and the coarea formula, Using (5.13) and definition of div n we easily get while passing to the limit in (5.12) as k → +∞ we deduce Passing to the limit in (5.15) as ε → 0 + we get lim sup so that the proof of (5.8) is complete. Finally, (5.9) follows from (5.14) and from (5.7) since div n is a measure, E r \E s E r \E r as s → r − .
We next investigate further regularity properties of V E .
In particular, (5.16) in the sense of distributions.
Proof We have, by coarea formula, from which the conclusion.

Corollary 5.5
For any t, r ∈ (0, +∞)\J with t < r we have In addition we have that for any r > ε > 0 As soon as r 0 is such that E r 0 ⊃ conv(E) ⊃ r ∈J (∂ E r ∩ E 1 r ) we have Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.