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 ballof center and radius in ; we will write for . will be the diskof center and radius in and ; will be the sphereof 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 fortioriSumming 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.
4. Examples
We first present the simplest case, of a compact subset .
Example 1 (Let ). For let be a monotone nonincreasing sequence, such that . Let and for . Consider a compact set is composed by countably many disjoint closed intervals spaced as in Figure 3, and the limit point is added. ThenLet , , define note that since . We can estimate for thatand indeed (setting for convenience) where the latter part is a disjoint union.
The example can be built in higher dimensions as well. Let for simplicity (the case is similar, by repeating spheres along the extra dimensions and changing some constants).
Example 2. Let be a monotone nondecreasing sequence of integers, such that and . Let . (Note that and ). Let and for . Consider a compact set union of a family of disks with centers in for , each of radiouses , as follows: It is easily proven thatLet then the compact set be is compact since we include in the line that contains limit points of . This set is tightly contained in the rectangle of corners and . See Figures 5 and 6 for two examples.
For any any two points in different families are at a distance ; moreover two different disks composing are at a distance at least (see Figure 4); henceLet , , and define (as before)note that since . Let . We can estimate for thatand indeed for the first inequality we note that when and , any two are at a distance . See Figure 4 again.
In general for the above examples will satisfy
We will need a lemma.
Lemma 12. Suppose that is continuous and decreasing and then . Equivalently (setting ), suppose that is continuous and increasing and and then .
Proof. Suppose instead that and let then and 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.
Proposition 13. Let be a continuous increasing function such that , , andand we will build a compact set, following the previous example, such that for
Proof. Let be the inverse , and then and . The subgraph coincides with the subgraph soLet ; note that . Since for and so ; so we can build as in the previous examples.
When we let as before; note that . We define as in (43) and note that is characterized by Since then and hence for Combining this with relation (46) we obtain (49).
When then we just set and proceed similarly.
4.1. Proof
We eventually prove Theorem 3.
Proof. Possibly adding a constant to and large disks to , we assume that . We relate for and ; we let and then is continuous and increasing, , , andso (48) is satisfied. By Lemma 12. We then build using the previous examples; by (55) and (46)
5. Alternative Proof
We now sketch a different proof of Theorem 1.
5.1. Standard Results
We will need these standard results in Geometric Measure Theory. In the following will be the characteristic function of a set .
Proposition 14 (Fleming–Rishel coarea formula [10]). Let be open and be locally integrable, and then where is the total mass of the distribution and is the perimeter of a Borel set inside .
(For a proof, see also Theorem 3.40 in [11]). We will write for .
Proposition 15 (Federer coarea formula [3]). Let be a Borel set and with Lipschitzian and integrable, and then In particular
Proposition 16. Let be an open set; let be Borel and . Let be the reduced boundary of and be the inward normal, and then
For definitions and proofs see, e.g., those in Section 3.5 in [11] or in Section 3 in [8].
Proposition 17. Let be an open set; let be Borel and . Let be the reduced boundary of , and then (i);(ii) for any 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.
Proposition 18 (structure theorem). If is a Borel set such that then for all and , where is the reduced boundary and is the inward normal. (For a definition, see also Theorem 3.54 in [11]).
5.2. Lemmas
Lemma 19. Let be open; suppose that is Lipschitz; let open. For almost all we have that
The proof follows comparing the Fleming–Rishel and the Federer coarea formula.
Lemma 20. Let be an integrable compactly supported function. Let be convex. Then the convolution is convex.
This is easily proved, e.g., by showing that the derivatives of are monotone nondecreasing.
Let be compact. It is well known that is semiconcave; we provide a simple quantitative proof.
Lemma 21. is concave.
Proof. Indeed but is affine hence is concave.
Lemma 22. Let again ; it is well known that (i) and are locally Lipschitz (by direct proof)(ii)hence for almost any the differentials exist,(iii)when these functions are differentiable and then(a) and ,(b)the point of closest to is unique and is ; for these last two, see in [13] (in particular Theorem 3.1 and Remark 3.6).
5.3. 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 . Let , supposing that the boundary of is smooth (in this case and we note that ; we have that for whereas 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 in the sense of distributions.
We now provide the general proof.
We sketch the following facts: (1)For almost all we have that , by the coarea formula. (Note that, by the proof in Section 3, this is actually true for every (2) by Proposition 9, so for any such that .(3)By Lemma 19 for almost any we have .(4)Hence for almost any and –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].(5)Let a family of mollifiers, with support in and . Let : then (possibly passing to a subsequence) for almost all we have Indeed we know that in (for any large), and we use the coarea formula and then (up to a subsequence) for almost all . Let be such that all above properties are satisfied. Let . 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 , , and thenBy Proposition 18the 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, is concave, and let by Lemma 20 is concave and smooth so then soThis ends the first part of the proof. Assuming that , we have proved the required result (2); namely, for almost any , i.e., for all where is a negligible set. Let then , and then there is a sequence with ; 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 of compact sets with smooth boundary such that . Indeed we may define 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.
5.4. 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 be a compact set; suppose that there are such thatThen for all where and is an appropriate positive constant dependent only on the dimension .
6. Banach-Like Metric Spaces of Compact Sets
Let be the family of all nonempty compact subsets of . In 2015 Duci and Mennucci [1] studied a family of distances on , where ([1] also includes the case which will not be considered here) and is a fixed given function. The distance was then defined, for , as
Lemma 24 (Lemma 6.2 in [1]). Suppose that is monotonically strictly decreasing andLet be closed and nonempty; then the following are equivalent: (a).(b) is bounded (and then is compact).
So, for any satisfying the above hypotheses, the formula in (85) would properly define a distance on .
6.1. Further Results on
Assuming appropriate hypotheses on , [1] proved many interesting properties of the metric space .
Hypotheses 25 (Hypotheses 6.1 in [1]). Suppose that is monotonically strictly decreasing and of class , such that
The resulting metric spaces enjoy many interesting properties.(i) is Euclidean invariant.(ii)The topology induced by over the space coincides with the topology induced by the Hausdorff distance (Theorem 6.11 in [1]).(iii)The metric space is complete (Theorem 6.12 in [1]).
6.2. Induced Distance and Geodesics
Given a metric space , we define the length of a continuous path by using the total variationwhere the supremum is computed over all finite subsets of and . We then define the induced distance bywhere 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 .
Hypotheses 26 (Hypotheses 6.17 in [1]). Let be as defined in Hypotheses 25. We moreover suppose that there is a constant such that is convex for , and
Then these results follow.(i)The space is Lipschitz–arc connected (Prop. 6.18 in [1]).(ii)For any , is compact in the topology (where is the distance induced by ) (Theorem 6.20 in [1]).(iii)Hence any two compact sets are connected by a geodesic in .(iv)When there is a variational description of the geodesics (Sec. 6.4 in [1]), a weak formulation of a “tangent bundle” (Sec. 6.5 in [1]).(v)For there is an interpretation of the metric as a “Riemannian metric” when smoothly deforming smooth boundaries of compact sets (Sec. 6.6 in [1]).
6.3. 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 or 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.
Proof. By Lemma 19 and Theorem 1 for almost any ,Let then be such that , and then ; so for almost all Let be as in hypotheses; we know (Lemma 22) that for almost all , and then by the Federer coarea formula Proposition 15 by relation (6), whereas by (7); so .
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
The author thanks Prof. Luigi Ambrosio (Scuola Normale Superiore) for the useful discussions. This work was supported by the PRIN 2015PA5MP7 (Calcolo delle Variazioni).