International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 8283496, 12 pages

https://doi.org/10.1155/2019/8283496

## On Perimeters and Volumes of Fattened Sets

Scuola Normale Superiore, Pisa, Italy

Correspondence should be addressed to Andrea C. G. Mennucci; ti.sns@iccunnem.aerdna

Received 21 November 2018; Accepted 17 January 2019; Published 24 April 2019

Academic Editor: Jean Michel Rakotoson

Copyright © 2019 Andrea C. G. Mennucci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper we analyze the shape of fattened sets; given a compact set let be its fattened set; we prove a general bound between the perimeter of and the Lebesgue measure of . We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, is integrable for . We further show that for any integrable continuous decreasing function there exists a compact set such that . These results solve a conjecture left open in (Mennucci and Duci, 2015) and provide new insight in applications where the fattened set plays an important role.

#### 1. Introduction

For any closed, let be the* distance function* from Let be the* fattened set* of of radious . It is equal to the Minkowski sum of A and a closed ball of radius r with center in the origin. It is also called* the parallel set* or the* tubolar neighborhood*.

Let be the -dimensional* Lebesgue measure*. Let be the* perimeter* of a Borel set .

##### 1.1. Main Results

In this short essay we will prove some geometrical properties of the function and of the fattened set.

The main result is as follows.

Theorem 1. *Let be a compact set. For all we have*

Note that (2) is sharp (consider the case ; see relations (16)).

We will provide two proofs of Theorem 1. An elementary proof is in Section 3; it is based on simple geometrical properties of sets in . Another proof is in Section 5, it is based on semiconcavity of and the Gauss–Green formula. It may be appreciated that the first method of proof is simpler.

A corollary of the above theorem is that, for any , the perimeter of the fattened set is finite—even when the perimeter of is not finite. We elaborate on this fact further.

*Remark 2. *Let for convenience; by the Fleming–Rishel coarea formula (see Proposition 14 and Lemma 24 here) (so the function is locally in ); then thesis (2) can be rewritten asthe above impliesfor .

The above properties (5) and (3) are again “sharp,” in this sense.

Theorem 3. *If is a continuous decreasing function satisfying and (that is requirement (3) in the above remark); then (as in (5)), and we can construct a compact set such that for .*

This holds in any (for ). The proof is in Section 4.1.

The interest on these results was spurred by the use of in [1]. We will discuss the connection to [1] in Section 6. The main theorem will be as follows.

Theorem 4. *Let and a Borel function such thatthen for any compact we have .*

Request (7) is justified by the existence of compact sets with properties as described in Theorem 3.

##### 1.2. Motivation

###### 1.2.1. Banach-Like Distances

Let be the family of all nonempty compact subsets of . In 2015 Duci and Mennucci [1] studied a family of distances on , defined by means of the distance function . Some natural questions came from that study; one of them was eventually answered by Theorem 4. See Section 6 for more details.

The study of fattened set though is quite wide and interesting; we provide two examples.

###### 1.2.2. Steiner Formulas

In 2004 Hug et al. [2] generalized the Steiner formulas as follows. Let . Fix closed, and let be the topological boundary.

*Definition 5. *Define the* normal bundle * as the set of pairs for which there exists a such that is the unique point of at minimum distance from (this definition is equivalent to the definition in Section 2.1 in [2], but it is simplified to avoid introducing further definitions and notations that are not needed in this paper); in that case, let be the supremum of such ; let for all .

is a Borel subset of . By Lemma 2.3 in [2], it is countably -rectifiable. There are examples of compact sets such that (the examples in this paper will do).

*Definition 6. *A* reach measure* is a real function with domain the Borel subsets of , such that (i) for all Borel subsets of (ii)at the same time, for any fixed , is a signed measure (of bounded variation) when evaluated on the family of Borel sets contained in

Theorem 7 (Theorem 2.1 in [2]). *For any closed set there exist reach measures such thatfor any and compact, and, for any bounded Borel,*

*Remark 8. *A closed set is* a set of positive reach* [3] when there exists a such that for any with there exists a unique at minimum distance from ; the* reach* is the largest such (possibly infinite). For example, smooth manifolds embedded in have positive reach, as well as convex subsets. It is easily verified that , where were defined in Definition 5. When the set has positive reach then for small formula (2) can be verified (through relation (3)) using results in [3]: see Theorem 5.5 in [3] that provides an explicit formula for , where is a polynomial in of degree at most .

In this respect it is worth noting this result by Fu et al [4]. For any that is a regular value for the distance function , the set is a set of positive reach (see Corollary 3.4 in [4]). When or a Sard–type result shows that the set of critical values is* small*, in an appropriate sense (see Thm. 4.1 in [4]).

###### 1.2.3. Minkowski Content

The study of the fattened set is linked to the Minkowski contentwhich has wide applications in the theory of Stochastic Differential Equations. See Ambrosio et al. [2, 5] and references therein.

#### 2. Notation

We will write for the* open ball*of center and radius in ; we will write for . will be the* disk*of center and radius in and ; will be the* sphere*of center and radius in .

For closed, let be the distance function from (defined in (1)). Let be the* fattened set* of of radious .

Let be the* Hausdorff distance* of compact sets in ; it can be defined (as shown in Sec. C in Chap. 4 in [6] and Sec. 2.2 in Chap. 4 in [7]) as

For let be the –dimensional* Hausdorff measure*; let be the –dimensional* Lebesgue measure*, and . Let be the* perimeter* of a Borel set , as defined in Definition 1.6 in [8]. We define and consequently for

#### 3. Area and Perimeter of Fattened Sets

These facts are known; see Sec. 4 and 5 in [1].

Proposition 9. *Let be a compact set. *(1)*Let . Let and for convenience.(i) The boundary of is contained in the set . (Equality may fail, consider (ii) is Lebesgue negligible (hence is as well).(iii)As a consequence, for any ,*(2)

*For any fixed , the fattening map is Lipschitz (of constant one) as a map from to .*(3)

*For any fixed , the*

*“fattened area map”*defined by is continuous on .*We add a further result in the same spirit.*

*Lemma 10. If are compact and according the Hausdorff distance and , are the fattened sets, then where is the symmetric difference of sets.*

*Proof. *Let again ; then for any we have that (i)if then for large (by (15)), so for large;(ii)if then for large, so for large;(iii)if then , that is negligible (by Proposition 9). The proof then follows from the Lebesgue dominated convergence theorem.

*We will also need this simple inequality.*

*Lemma 11. For each and integer we have*

*Proof. *We have

*We now come to the proof of the main contribution of this paper, Theorem 1.*

*Proof. *The proof is in three steps. (1)Let and two disjoint disks (as defined in (13)). The locus of points at the same distance from the two disks is (a connected sheet of) a quadric. Choose appropriate coordinates where , , with , and so that the locus contains the origin; then the locus can be written as See Figure 1 Suppose that is the union of finitely many disjoint disks. Let for simplicity be the distance to one such disk. Note that Fix and then consider the region of points that are nearer to than to any other disk; is usually called a* Voronoi cell*. is defined by the inequalities These can be reduced to inequalities involving first and second-degree polynomials in . For example, the three inequalities when can be reduced (in appropriate coordinates as above, setting ) toSo region is an open semialgebraic set. Its boundary is a semialgebraic set. It is contained in the finite union of quadrics as described in (20).

The set is contained in the union of spheres . A sphere can intersect a quadric such as (20) in a set of at most dimension . So when evaluating we only consider the parts of that are inside the regions .

Inside each region the set is given by the equality , so it is a part of a sphere. For each point , the projection of to is an unique point contained in . The segment from to passes through , and it is contained in . This follows from well-known theory for distance functions, but in this case can also be easily checked with simple geometrical arguments. We denote by the union of all segments for . See Figure 2.

So we can establish the relations ( is the solid angle under which sees ). By Lemma 11 then Since then* a fortiori*Summing in we obtain the relationThat is the same as (2) for the case when is an union of spheres.(2)Let be a compact set. Define : this is a decreasing sequence of open sets such that , , , and .Fix ; by Vitali’s covering theorem we can choose finitely many disjoint disks inside so that their union is , and it satisfies ; we also have that .

Summarizing we obtain a sequence such that each is the union of finitely many disjoint disks, and .(3)Since then by Lemma 10 Using moreover the fact that then Since then, by Thm. 1.9 in [8], So (2) follows.