On Perimeters and Volumes of Fattened Sets

In this paper we analyze the shape of fattened sets; given a compact set 𝐶 ⊂ R 𝑁 let 𝐶 𝑟 be its 𝑟− fattened set; we prove a general bound 𝑟𝑃(𝐶 𝑟 ) ≤ 𝑁 L ({𝐶 𝑟 \ 𝐶}) between the perimeter of 𝐶 𝑟 and the Lebesgue measure of 𝐶 𝑟 \ 𝐶 . We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, 𝑃(𝐶 𝑟 ) is integrable for 𝑟 ∈ (0,𝑎) . We further show that for any integrable continuous decreasing function 𝜓 : (0, 1/2) 󳨀→ (0,∞) there exists a compact set 𝐶 ⊂ R 𝑁 such that 𝑃(𝐶 𝑟 ) ≥ 𝜓(𝑟) . 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.


Introduction
For any ⊆ R closed, let be the distance function from Let { ≤ } = { ∈ R : ( ) ≤ } be the fattened set of of radious > 0. It is equal to the Minkowski sum of A and a closed ball (0) of radius r with center in the origin. It is also called the parallel set or the tubolar neighborhood. Let L be the -dimensional Lebesgue measure. Let ( ) be the perimeter of a Borel set ⊆ R .

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.
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 R . Another proof is in Section 5, it is based on semiconcavity of 2 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 > 0, the perimeter of the fattened set { ≤ } is finite-even when the perimeter of is not finite. We elaborate on this fact further.
This holds in any R (for ≥ 1). 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.
then for any compact ⊆ R we have ∘ ∈ (R ).
Request (7) is justified by the existence of compact sets with properties as described in Theorem 3. Definition 5. Define the normal bundle ( ) ⊆ × 1 as the set of pairs ( , ) ∈ × 1 for which there exists a > 0 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 ( , ) = 0 for all ( , ) ∈ (R × 1 ) \ ( ).

Definition 6.
A reach measure is a real function with domain the Borel subsets of R × 1 , such that (i) ( ) = 0 for all Borel subsets of (R × 1 ) \ ( ) (ii) at the same time, for any fixed > 0, ( ) 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 0 , 1 . . . −1 such that for any > 0 and ⊆ R compact, and, for any : Remark 8. A closed set is a set of positive reach [3] when there exists a > 0 such that for any ∈ R with ( ) < there exists a unique ∈ at minimum distance from ; the reach is the largest such (possibly infinite). For example, smooth manifolds embedded in R have positive reach, as well as convex subsets. It is easily verified that = inf{ ( , ) : ( , ) ∈ ( )}, where ( ), were defined in Definition 5. When the set has positive reach then for small > 0 formula (2) can be verified (through relation (3)) using results in [3]: see Theorem 5.5 in [3] that provides an explicit formula for L({ ≤ }) = ( ), 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 > 0 that is a regular value for the distance function , the set R \ is a set of positive reach (see Corollary 3.4 in [4]). When = 2 or = 3 a Sard-type result shows that the set of critical values is small, in an appropriate sense (see Thm. 4.1 in [4]).

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

Notation
We will write ( ) for the open ball International Journal of Mathematics and Mathematical Sciences 3 of center and radius > 0 in R ; we will write for (0). ( ) will be the disk of center and radius > 0 in R and = (0); ( ) will be the sphere of center and radius > 0 in R . For ⊆ R closed, let be the distance function from (defined in (1)     We add a further result in the same spirit. (ii) if ( ) < then ( ) < for large, so ∉̃Δf or large; The proof then follows from the Lebesgue dominated convergence theorem.
We will also need this simple inequality.
Lemma 11. For each , ≥ 0 and ≥ 1 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 , ∈ R , , > 0 and ( ), ( ) two disjoint disks (as defined in (13)). The locus of points ∈ R at the same distance from the two disks is (a connected sheet of) a quadric. Choose appropriate coordinates where = (− , 0, . . . , 0), = ( , 0, . . . , 0), with 0 < < , 0 < < < and − = − so that the locus contains the origin; then the locus can be written as See Figure 1 Suppose that = ⋃ =1 ( ) is the union of finitely many disjoint disks. Let for simplicity z w Figure 1 be the distance to one such disk. Note that Fix ∈ 1, . . . 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 = ( 1 , . . . ). For example, the three inequalities when < can be reduced (in appropriate coordinates as above, setting = , = , = , = ) to So 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 ⋃ =1 + ( ). A sphere can intersect a quadric such as (20) in a set of at most dimension − 2. So when evaluating H −1 ({ = }) 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 ( > 0 is the solid angle under which sees { = } ∩ ). By Lemma 11 then Since then a fortiori Summing in we obtain the relation That is the same as (2) for the case when is an union of spheres.
(2) Let be a compact set. Define Fix ; by Vitali's covering theorem we can choose finitely many disjoint disks inside so that their union is , and it satisfies L( \ ) ≤ 1/ ; we also have that d ( , ) ≤ 1/ .

Examples
We first present the simplest case, of a compact subset ⊆ R.
is composed by countably many disjoint closed intervals spaced as in Figure 3, and the limit point { ∞ } is added. Then note that ( ) < ∞ since lim ℎ →∞ ℎ = 0. We can estimate for < 1 that and indeed (setting = ( ) for convenience) where the latter part is a disjoint union.
The example can be built in higher dimensions as well. Let = 2 for simplicity (the case ≥ 3 is similar, by repeating spheres along the extra dimensions and changing some constants).
In general for ≥ 1 the above examples will satisfy We will need a lemma. Proof. Suppose instead that lim sup →∞ ( )/ > 0 and let then > 0 and > 0 be an increasing sequence such that → ∞ and for all we have ( ) > : then where the last step is explained in exercise 11 in Section 3 in [9].
The above example can be finetuned as follows.

Alternative Proof
We now sketch a different proof of Theorem 1.

Standard Results.
We will need these standard results in Geometric Measure Theory. In the following 1 will be the characteristic function of a set .
Proposition 14 (Fleming-Rishel coarea formula [10]). Let Ω ⊆ R be open and : Ω → R be locally integrable, and then is the total mass of the distribution and is the perimeter of a Borel set inside Ω.

(63)
In particular For definitions and proofs see, e.g., those in Section 3.5 in [11] or in Section 3 in [8].

Proposition 17. Let Ω ⊆ R be an open set; let be Borel and
( , Ω) < ∞. Let * be the reduced boundary of , and then for any ⊆ Ω open.
These results are due to De Giorgi [12]; for a proof see, e.g., in Section 3.5 in [11] or in Section 4 in [8]. Combining the above with the Fleming-Rishel coarea formula we obtain the following.
The proof follows comparing the Fleming-Rishel and the Federer coarea formula.
Let be compact. It is well known that 2 is semiconcave; we provide a simple quantitative proof.
is affine hence is concave. for these last two, see in [13] (in particular Theorem 3.1 and Remark 3.6).

Proof.
Here is then a "geometric measure theory proof" of Theorem 1.
Proof. In the first part of this proof we assume that ( ) < ∞. Let = { ≤ } and = { = } in the following. The idea of the proof is as follows. Set ( ) = 2 ( ). Let = {0 < ≤ } = \ , supposing that the boundary of is smooth (in this case = .), and we note that = ∪ ; we have that ∇ ( ) = 0 for ∈ whereas ∇ ( ) = −2 ( )] = −2 ]( ) for ∈ , where ]( ) is the inward normal to ; so we use the Gauss-Green formula and write where the first inequality follows from Lemma 21 that implies that Δ( 2 ) ≤ 2 in the sense of distributions.
We now provide the general proof. We sketch the following facts: (1) For almost all > 0 we have that ( ) < ∞, by the coarea formula. (Note that, by the proof in Section 3, this is actually true for every > 0.) (2) ⊆ by Proposition 9, so * ⊆ for any > 0 such that ( ) < ∞.
(4) Hence for almost any > 0 and H −1 -almost any ∈ we have that that is the outward normal. This follows from the existence of a "weak" tangent space of * near ; see Theorem 3.8 in [8].
Let > 0 be such that all above properties are satisfied. Let = {0 < ≤ } = \ . Note that ( ) < ∞ (since the boundaries of and are separated); and the essential boundary of is the union of the essential boundaries of and of . For any point in the topological boundary of , lim → ∇ ( ) = 0, and then By Proposition 18 the right hand side passes to the limit (by (75) and by (73)) For the left hand side of (76) we proceed as follows. By Lemma 21, ( ) = ( ) − | | 2 is concave, and let by Lemma 20 ℎ is concave and smooth so ∇ ℎ ≤ 0 This ends the first part of the proof. Assuming that ( ) < ∞, we have proved the required result (2); namely, for almost any > 0, i.e., for all > 0, ∉ where ⊆ R is a negligible set. Let then > 0, ∈ , and then there is a sequence ℎ ↘ ℎ with ℎ ∉ ; by Proposition 9 (point (2)) and the lower semicontinuity of the perimeter we deduce that (2) holds for as well.
We eventually extend (2) to the general case. Suppose is compact and ( ) = ∞ and then there is a sequence ℎ of compact sets with smooth boundary such that ( ℎ , ) → ℎ 0. Indeed we may define ℎ = { : Φ ℎ * 1 ( ) ≥ } where is a regular value and then use Sard's theorem. By Lemma 10 and the lower semicontinuity of the perimeter we deduce that (2) holds.

Almgren Taylor Wang.
It is worth mentioning that the above proof follows closely part of the proof by Ambrosio (lectures notes Minimizing Movements held in Padova, June 20-24, 1994) of this result by Almgren, Taylor, and Wang [14].
Theorem 23. Let ⊂ R be a compact set; suppose that there are > 0, > 0 such that Then for all > ess-sup where = −1 1− and > 0 is an appropriate positive constant dependent only on the dimension .

Banach-Like Metric Spaces of Compact Sets
Let M be the family of all nonempty compact subsets of R . In 2015 Duci and Mennucci [1] studied a family of distances , on M, where ∈ [1, ∞) ([1] also includes the case = ∞, which will not be considered here) and : [0, ∞) → (0, ∞) is a fixed given function. The distance was then defined, for , ∈ M, as Lemma 24 (Lemma 6.2 in [1]). Suppose that is monotonically strictly decreasing and Let ⊆ R be closed and nonempty; then the following are equivalent: (a) ∘ ∈ (R ).

(b) is bounded (and then is compact).
So, for any satisfying the above hypotheses, the formula in (85) would properly define a distance on M. Hypotheses 25 (Hypotheses 6.1 in [1]). Suppose that is monotonically strictly decreasing and of class 1 , such that The resulting metric spaces enjoy many interesting properties.
(ii) The topology induced by , over the space M coincides with the topology induced by the Hausdorff distance (Theorem 6.11 in [1]). (iii) The metric space (M, , ) is complete (Theorem 6.12 in [1]).

Induced Distance and Geodesics.
Given a metric space ( , ), we define the lengthLen of a continuous path : [ , ] → by using the total variation where the supremum is computed over all finite subsets = { 0 , . . . , } of [ , ] and 0 ≤ ⋅ ⋅ ⋅ ≤ . We then define the induced distance by where the infimum is taken in the class of all continuous paths connecting to . If the infimum is a minimum, the path providing the minimum is called a . With further hypotheses on ( ), more can be said on the metric space (M, , ).
Then these results follow.
(iii) Hence any two compact sets are connected by a geodesic in (M, , ).
(v) For = 2 there is an interpretation of the metric , as a "Riemannian metric" when smoothly deforming smooth boundaries of compact sets (Sec. 6.6 in [1]).

Open Problem.
All the above properties are quite interesting for their applications in Shape Analysis and Optimization. In this context, the set of Hypotheses 25 and Hypotheses 26 can be satisfied by simple functions such as ( ) = exp(− ) or ( ) = (1 + ) − for > / . From a theoretical point of view, it may be interesting instead to understand more in detail which hypotheses are strictly needed to prove each of the above statements. This requires a deeper understanding of the properties of the distance function and the fattened sets. The theorems in this paper are a step in this direction.
In proving Lemma 24 it was assumed that be monotonically strictly decreasing. This was a useful ingredient in writing a simple direct proof of the implication ⇒ . Some geometrical and analytical considerations though suggested that this hypothesis was not strictly needed.
Indeed as a consequence of Theorem 1 we can now prove Theorem 4.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares no conflicts of interest.