# On Perimeters and Volumes of Fattened Sets

Authors: Andrea C. G. Mennucci
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

In this paper we analyze the shape of fattened sets; given a compact setC⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C. We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, P(Cr) 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⊂RN such that P(Cr)≥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.


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## Body

## 1. Introduction

For anyA⊆RN closed, let uA be thedistance function from A(1)uAx=definfy∈A⁡y-x.Let {uA≤r}={x∈RN:uA(x)≤r} be thefattened set of A of radious r>0. It is equal to the Minkowski sum of A and a closed ball Dr(0) of radius r with center in the origin. It is also calledthe parallel set or thetubolar neighborhood.LetL be the N-dimensionalLebesgue measure. Let P(E) be theperimeter of a Borel set E⊆RN.


### 1.1. Main Results

In this short essay we will prove some geometrical properties of the functionuA and of the fattened set.The main result is as follows.Theorem 1.
LetC⊂RN be a compact set. For all r>0 we have(2)rPuC≤r≤NL0<uC≤r.Note that (2) is sharp (consider the case C={0}; see relations (16)).We will provide two proofs of Theorem1. An elementary proof is in Section 3; it is based on simple geometrical properties of sets in RN. Another proof is in Section 5, it is based on semiconcavity of uC2 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 anyr>0, the perimeter of the fattened set {uC≤r} is finite—even when the perimeter of C is not finite. We elaborate on this fact further.Remark 2.
Letψ(r)=P({uC≤r}) for convenience; by the Fleming–Rishel coarea formula (see Proposition 14 and Lemma 24 here)(3)L0<uC≤r=∫0rψsds<∞ (so the function ψ is locally in L1); then thesis (2) can be rewritten as(4)ψr≤Nr∫0rψsds;the above implies(5)ψr=o1rfor r→0.The above properties (5) and (3) are again “sharp,” in this sense.Theorem 3.
Ifψ:(0,1/2)→(0,∞) is a continuous decreasing function satisfying limr→0⁡ψ(r)=∞ and ∫01/2ψ(s)ds<∞ (that is requirement (3) in the above remark); then ψ(r)=o(1/r) (as in (5)), and we can construct a compact set C such that P({uC≤r})≥ψ(r) for r∈(0,1/2).This holds in anyRN (for N≥1). The proof is in Section 4.1.The interest on these results was spurred by the use ofuA in [1]. We will discuss the connection to [1] in Section 6. The main theorem will be as follows.Theorem 4.
Letp∈[1,∞) and φ:[0,∞)→[0,∞) a Borel function such that(6)∫0∞tN-1φtpdt<∞,(7)∃ε>0,ess-sup0<t<ε⁡φt<∞;then for any compact C⊆RN we have φ∘uC∈Lp(RN).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

LetM be the family of all nonempty compact subsets of RN. In 2015 Duci and Mennucci [1] studied a family of distances on M, defined by means of the distance function uA. 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 S1={u∈RN:|u|=1}. Fix C⊆RN closed, and let ∂C be the topological boundary.Definition 5.
Define thenormal bundleN(C)⊆C×S1 as the set of pairs (x,u)∈C×S1 for which there exists a t>0 such that x is the unique point of C at minimum distance from x+tu (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 δ(x,u) be the supremum of such t; let δ(x,u)=0 for all (x,u)∈(RN×S1)∖N(C).N ( C ) is a Borel subset of (∂C)×S1. By Lemma 2.3 in [2], it is countably (N-1)-rectifiable. There are examples of compact sets such that HN-1(N(C))=∞ (the examples in this paper will do).Definition 6.
Areach measure is a real function μ with domain the Borel subsets of RN×S1, such that (i)
μ(A)=0 for all Borel subsets of (RN×S1)∖N(C)(ii)
at the same time, for any fixeda>0, μ(A) is a signed measure (of bounded variation) when evaluated on the family of Borel sets A contained in(8)x,u∈NC:x≤a,δx,u≥1a.Theorem 7 (Theorem 2.1 in [2]).
For any closed setC there exist reach measures μ0,μ1…μN-1 such that(9)∫x∈B,u∈S1δx,u∧rN-jdμjx,u<∞for any r>0 and B⊆RN compact, and, for any f:RN→R bounded Borel,(10)∫RN∖Cfxdx=∑i=0N-1N-iωN-i∫NC∫0δx,utN-1-ifx+tudtdμix,uRemark 8.
A closed setC isa set of positive reach [3] when there exists a ρ>0 such that for any x∈RN with uC(x)<ρ there exists a unique y∈C at minimum distance from x; thereach is the largest such ρ (possibly infinite). For example, smooth manifolds embedded in RN have positive reach, as well as convex subsets. It is easily verified that ρ=inf⁡{δ(x,u):(x,u)∈N(C)}, where N(C),δ were defined in Definition 5. When the set C has positive reach then for small r>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({uC≤r})=p(r), where p(r) is a polynomial in r of degree at most N.
In this respect it is worth noting this result by Fu et al [4]. For any r>0 that is a regular value for the distance function uC, the set RN∖Cr¯ is a set of positive reach (see Corollary 3.4 in [4]). When N=2 or N=3 a Sard–type result shows that the set of critical values issmall, 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 content(11)limr→0⁡LuA≤rrwhich has wide applications in the theory of Stochastic Differential Equations. See Ambrosio et al. [2, 5] and references therein.


## 1.1. Main Results

In this short essay we will prove some geometrical properties of the functionuA and of the fattened set.The main result is as follows.Theorem 1.
LetC⊂RN be a compact set. For all r>0 we have(2)rPuC≤r≤NL0<uC≤r.Note that (2) is sharp (consider the case C={0}; see relations (16)).We will provide two proofs of Theorem1. An elementary proof is in Section 3; it is based on simple geometrical properties of sets in RN. Another proof is in Section 5, it is based on semiconcavity of uC2 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 anyr>0, the perimeter of the fattened set {uC≤r} is finite—even when the perimeter of C is not finite. We elaborate on this fact further.Remark 2.
Letψ(r)=P({uC≤r}) for convenience; by the Fleming–Rishel coarea formula (see Proposition 14 and Lemma 24 here)(3)L0<uC≤r=∫0rψsds<∞ (so the function ψ is locally in L1); then thesis (2) can be rewritten as(4)ψr≤Nr∫0rψsds;the above implies(5)ψr=o1rfor r→0.The above properties (5) and (3) are again “sharp,” in this sense.Theorem 3.
Ifψ:(0,1/2)→(0,∞) is a continuous decreasing function satisfying limr→0⁡ψ(r)=∞ and ∫01/2ψ(s)ds<∞ (that is requirement (3) in the above remark); then ψ(r)=o(1/r) (as in (5)), and we can construct a compact set C such that P({uC≤r})≥ψ(r) for r∈(0,1/2).This holds in anyRN (for N≥1). The proof is in Section 4.1.The interest on these results was spurred by the use ofuA in [1]. We will discuss the connection to [1] in Section 6. The main theorem will be as follows.Theorem 4.
Letp∈[1,∞) and φ:[0,∞)→[0,∞) a Borel function such that(6)∫0∞tN-1φtpdt<∞,(7)∃ε>0,ess-sup0<t<ε⁡φt<∞;then for any compact C⊆RN we have φ∘uC∈Lp(RN).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

LetM be the family of all nonempty compact subsets of RN. In 2015 Duci and Mennucci [1] studied a family of distances on M, defined by means of the distance function uA. 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 S1={u∈RN:|u|=1}. Fix C⊆RN closed, and let ∂C be the topological boundary.Definition 5.
Define thenormal bundleN(C)⊆C×S1 as the set of pairs (x,u)∈C×S1 for which there exists a t>0 such that x is the unique point of C at minimum distance from x+tu (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 δ(x,u) be the supremum of such t; let δ(x,u)=0 for all (x,u)∈(RN×S1)∖N(C).N ( C ) is a Borel subset of (∂C)×S1. By Lemma 2.3 in [2], it is countably (N-1)-rectifiable. There are examples of compact sets such that HN-1(N(C))=∞ (the examples in this paper will do).Definition 6.
Areach measure is a real function μ with domain the Borel subsets of RN×S1, such that (i)
μ(A)=0 for all Borel subsets of (RN×S1)∖N(C)(ii)
at the same time, for any fixeda>0, μ(A) is a signed measure (of bounded variation) when evaluated on the family of Borel sets A contained in(8)x,u∈NC:x≤a,δx,u≥1a.Theorem 7 (Theorem 2.1 in [2]).
For any closed setC there exist reach measures μ0,μ1…μN-1 such that(9)∫x∈B,u∈S1δx,u∧rN-jdμjx,u<∞for any r>0 and B⊆RN compact, and, for any f:RN→R bounded Borel,(10)∫RN∖Cfxdx=∑i=0N-1N-iωN-i∫NC∫0δx,utN-1-ifx+tudtdμix,uRemark 8.
A closed setC isa set of positive reach [3] when there exists a ρ>0 such that for any x∈RN with uC(x)<ρ there exists a unique y∈C at minimum distance from x; thereach is the largest such ρ (possibly infinite). For example, smooth manifolds embedded in RN have positive reach, as well as convex subsets. It is easily verified that ρ=inf⁡{δ(x,u):(x,u)∈N(C)}, where N(C),δ were defined in Definition 5. When the set C has positive reach then for small r>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({uC≤r})=p(r), where p(r) is a polynomial in r of degree at most N.
In this respect it is worth noting this result by Fu et al [4]. For any r>0 that is a regular value for the distance function uC, the set RN∖Cr¯ is a set of positive reach (see Corollary 3.4 in [4]). When N=2 or N=3 a Sard–type result shows that the set of critical values issmall, 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 content(11)limr→0⁡LuA≤rrwhich has wide applications in the theory of Stochastic Differential Equations. See Ambrosio et al. [2, 5] and references therein.


## 1.2.1. Banach-Like Distances

LetM be the family of all nonempty compact subsets of RN. In 2015 Duci and Mennucci [1] studied a family of distances on M, defined by means of the distance function uA. 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 S1={u∈RN:|u|=1}. Fix C⊆RN closed, and let ∂C be the topological boundary.Definition 5.
Define thenormal bundleN(C)⊆C×S1 as the set of pairs (x,u)∈C×S1 for which there exists a t>0 such that x is the unique point of C at minimum distance from x+tu (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 δ(x,u) be the supremum of such t; let δ(x,u)=0 for all (x,u)∈(RN×S1)∖N(C).N ( C ) is a Borel subset of (∂C)×S1. By Lemma 2.3 in [2], it is countably (N-1)-rectifiable. There are examples of compact sets such that HN-1(N(C))=∞ (the examples in this paper will do).Definition 6.
Areach measure is a real function μ with domain the Borel subsets of RN×S1, such that (i)
μ(A)=0 for all Borel subsets of (RN×S1)∖N(C)(ii)
at the same time, for any fixeda>0, μ(A) is a signed measure (of bounded variation) when evaluated on the family of Borel sets A contained in(8)x,u∈NC:x≤a,δx,u≥1a.Theorem 7 (Theorem 2.1 in [2]).
For any closed setC there exist reach measures μ0,μ1…μN-1 such that(9)∫x∈B,u∈S1δx,u∧rN-jdμjx,u<∞for any r>0 and B⊆RN compact, and, for any f:RN→R bounded Borel,(10)∫RN∖Cfxdx=∑i=0N-1N-iωN-i∫NC∫0δx,utN-1-ifx+tudtdμix,uRemark 8.
A closed setC isa set of positive reach [3] when there exists a ρ>0 such that for any x∈RN with uC(x)<ρ there exists a unique y∈C at minimum distance from x; thereach is the largest such ρ (possibly infinite). For example, smooth manifolds embedded in RN have positive reach, as well as convex subsets. It is easily verified that ρ=inf⁡{δ(x,u):(x,u)∈N(C)}, where N(C),δ were defined in Definition 5. When the set C has positive reach then for small r>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({uC≤r})=p(r), where p(r) is a polynomial in r of degree at most N.
In this respect it is worth noting this result by Fu et al [4]. For any r>0 that is a regular value for the distance function uC, the set RN∖Cr¯ is a set of positive reach (see Corollary 3.4 in [4]). When N=2 or N=3 a Sard–type result shows that the set of critical values issmall, 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 content(11)limr→0⁡LuA≤rrwhich has wide applications in the theory of Stochastic Differential Equations. See Ambrosio et al. [2, 5] and references therein.


## 2. Notation

We will writeBr(x) for theopen ball(12)Brx=defy∈RN:x-y<rof center x and radius r>0 in RN; we will write Br for Br(0). Dr(x) will be thedisk(13)Drx=defy∈RN:x-y≤rof center x and radius r>0 in RN and Dr=Dr(0); Sr(x) will be thesphere(14)Srx=defy∈RN:x-y=rof center x and radius r>0 in RN.ForA⊆RN closed, let uA be the distance function from A (defined in (1)). Let {uA≤r} be thefattened set of A of radious r>0.LetdH be theHausdorff distance of compact sets in RN; it can be defined (as shown in Sec. C in Chap. 4 in [6] and Sec. 2.2 in Chap. 4 in [7]) as(15)dHA,B=defsupx∈RN⁡uAx-uBx.Ford≥0 let Hd be the d–dimensionalHausdorff measure; let L be the N–dimensionalLebesgue measure, and ωN=defL(B1). Let P(E) be theperimeter of a Borel set E⊆RN, as defined in Definition 1.6 in [8]. We define ωN=L(B1) and consequently for r>0(16)LBr=ωNrN,PBr=HN-1Sr=NωNrN-1,LBr=ωNrN=rNHN-1Sr.


## 3. Area and Perimeter of Fattened Sets

These facts are known; see Sec. 4 and 5 in [1].Proposition 9.
LetA⊂RN be a compact set. (1)
Letr>0. Let F={uA≤r} and E={uA=r} for convenience.
(i)
The boundary∂F of F is contained in the set E. (Equality may fail, consider A={x∈[1,2]},r=1.)(ii)
E is Lebesgue negligible (hence ∂F is as well).(iii)
As a consequence, for anyr>0,(17)PuC≤r=PuC<r.(2)
For any fixedA∈M, the fattening map λ↦{uA≤λ} is Lipschitz (of constant one) as a map from [0,∞) to (M,dH).(3)
For any fixedλ>0, the“fattened area map”Lλ:M→R defined by Lλ(A)=defL({uA≤λ}) is continuous on (M,dH).We add a further result in the same spirit.Lemma 10.
IfAn,A are compact and An→A according the Hausdorff distance and A~={uA≤λ}, A~n={uAn≤λ} are the fattened sets, then L(A~nΔA~)→0 where Δ is the symmetric difference of sets.Proof.
Let againE={x:uA(x)=λ}; then for any x∈A~nΔA~ we have that (i)
ifuA(x)>λ then uAn(x)>λ for n large (by (15)), so x∉A~nΔA~ for n large;(ii)
ifuA(x)<λ then uAn(x)<λ for n large, so x∉A~nΔA~ for n large;(iii)
ifuA(x)=λ then x∈E, 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 eachr,R≥0 and N≥1 integer we have(18)r+RN-RN≥rr+RN-1.Proof.
We have(19)r+RN-RN=∑j=0N-1NjrN-jRj,rr+RN-1=∑j=0N-1N-1jrN-jRj,NjN-1j=NN-j≥1∀j=0,…,N-1.We now come to the proof of the main contribution of this paper, Theorem1.Proof.
The proof is in three steps.(1)
Letz,w∈RN,r,ρ>0 and Dr(z),Dρ(w) two disjoint disks (as defined in (13)). The locus of points x∈Rn at the same distance from the two disks is (a connected sheet of) a quadric. Choose appropriate coordinates where z=(-a,0,…,0), w=(b,0,…,0), with 0<r<a, 0<ρ<b<a and a-r=b-ρ so that the locus contains the origin; then the locus can be written as(20)x1≥0∧4abx1x1+a-b=a-b2∑j=2nxj2.See Figure1Suppose thatC=⋃i=1nDri(zi) is the union of finitely many disjoint disks. Let for simplicity(21)δix=uDrizix=x-zi-ri+ be the distance to one such disk. Note that(22)uCx=mini=1,…,n⁡δix. Fix k∈1,…n and then consider the region Rk of points x that are nearer to Drk(zk) than to any other disk; Rk is usually called aVoronoi cell. Rk is defined by the inequalities(23)Rk=x:∀j≠k,δkx<δjx. These can be reduced to inequalities involving first and second-degree polynomials in x=(x1,…xn). For example, the three inequalities(24)x-zj>rj,x-zk>rk,δkx<δjx when rk<rj can be reduced (in appropriate coordinates as above, setting z=zk,w=zj,ρ=rk,r=rj) to(25)x1-b2+∑j=2nxj2>ρ2∧x1>0∧4abx1x1+a-b>a-b2∑j=2nxj2.So region Rk 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{uC=r} is contained in the union of spheres ⋃i=1nSr+ri(zi). A sphere can intersect a quadric such as (20) in a set of at most dimension N-2. So when evaluating HN-1({uC=r}) we only consider the parts of {uC=r} that are inside the regions Rk.
Inside each regionRk the set {uC=r} is given by the equality x-zi=r+rk, so it is a part of a sphere. For each point x∈{uC=r}∩Rk, the projection of x to C is an unique point p contained in Srk(zk). The segment from x to zk passes through p, and it is contained in Rk. 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 J the union of all segments xzk for x∈{uC=r}∩Rk. See Figure 2.
So we can establish the relations(26)LJ∖C=αNr+rkN-rkN,HN-1uC=r∩Rk=r+rkN-1α. (α>0 is the solid angle under which zk sees {uC=r}∩Rk). By Lemma 11 then(27)rHN-1uC=r∩Rk≤NLJ∖C. Since(28)J∖C⊆0<uC≤r∩Rk thena fortiori(29)rHN-1uC=r∩Rk≤NL0<uC≤r∩Rk.
Summing ink we obtain the relation(30)rHN-1uC=r≤NL0<uC≤r.That is the same as (2) for the case when C is an union of spheres.(2)
LetC be a compact set. Define An={uC<1/n}: this is a decreasing sequence of open sets An such that An⊇C, ⋂n=1∞An=C, L(An)→nL(A), and dH(An,C)≤1/n.
Fixn; by Vitali’s covering theorem we can choose finitely many disjoint disks inside An so that their union is Cn, and it satisfies L(An∖Cn)≤1/n; we also have that dH(An,Cn)≤1/n.
Summarizing we obtain a sequenceCn such that each Cn is the union of finitely many disjoint disks, dH(Cn,C)→n0 and L(CΔCn)→n0.(3)
SincedH(Cn,C)→0 then by Lemma 10(31)LuCn≤rΔuC≤r→0.Using moreover the fact thatL(CΔCn)→n0 then(32)L0<uCn≤rΔ0<uC≤r→0.SinceL(CΔCn)→n0 then, by Thm. 1.9 in [8], (33)PC≤liminfn⁡PCn.So (2) follows.Figure 1Figure 2
Voronoi cells, regions.


## 4. Examples

We first present the simplest case, of a compact subsetC⊆R.Example 1 (LetN=1).
Forh∈N,h≥1 let rh>0 be a monotone nonincreasing sequence, such that β=∑h=1∞rh<∞. Let a1=0 and ah=4∑j=1h-1rj for h≥2. Consider a compact set(34)C=a∞∪⋃h=1∞ah-rh,ah+rh.
C is composed by countably many disjoint closed intervals spaced as in Figure 3, and the limit point {a∞} is added. Then(35)LC=2β<∞,PC=∞.
Letr>0, r<r1, define(36)χr=defmax⁡h≥1:r<rh; note that χ(r)<∞ since limh→∞rh=0. We can estimate for r<r1 that(37)PuC≤r≥2χr,and indeed (setting k=χ(r) for convenience)(38)uC≤r∩-∞,ak=ak-rk-r,ak∪⋃h=1k-1ah-rh-r,ah+rh+r, where the latter part is a disjoint union.Figure 3The example can be built in higher dimensions as well. LetN=2 for simplicity (the case N≥3 is similar, by repeating spheres along the extra dimensions and changing some constants).Example 2.
Let(mh)h≥1 be a monotone nondecreasing sequence of integers, such that mh≥1 and ∑h=1∞1/mh<∞. Let rh=4/(4mh-2). (Note that 2≥rhmh≥1 and ∑h=1∞rh<∞). Let a1=r1 and ah=a1+4∑j=1h-1rj for h≥2. Consider a compact set Ch union of a family of mh disks with centers in xh,j=(ah,rh(1+4j)) for j=0,1,…,mh-1, each of radiouses rh, as follows:(39)Ch=def⋃j=0mh-1Dxh,j,rh. It is easily proven that(40)LCh=πrh2mh≤2πrh,PCh=2πrhmh≥2π.
Let then the compact set be(41)C=defa∞,t:t∈0,4∪⋃h≥1ChC is compact since we include in C the line that contains limit points of ⋃h≥1Ch. This set C is tightly contained in the rectangle of corners (0,0) and (a∞,4). See Figures 5 and 6 for two examples.
For anyh<k any two points x∈Ch,y∈Ck in different families are at a distance x-y≥2rh; moreover two different disks composing Ch are at a distance at least 2rh (see Figure 4); hence(42)LC<∞,PC=∞.
Letr>0, r<r1, and define (as before)(43)χr=defmax⁡h≥1:r<rh;note that χ(r)<∞ since limh→∞⁡rh=0. Let Mh=∑j=1hmj. We can estimate for r<r1 that(44)PuC≤r≥2π∑h=1χrr+rhmh≥2πχr,and indeed for the first inequality we note that when r<rh and k>h, any two(45)x∈uCk≤r,y∈uCh≤r are at a distance |x-y|≥2(rh-r)>0. See Figure 4 again.Figure 4
Distances between disks inCh,Ch+1.Figure 5
Compact setC (in gray) from Example 2 with mh=⌈(3/2)h⌉. (The blue, dotted rectangle has corners (0,0) and (a∞,4) and is not part of C.)Figure 6
Compact setC (in gray) from Example 2 with mh=1+h2. (The blue, dotted rectangle has corners (0,0) and (a∞,4) and is not part of C.)In general forN≥1 the above examples will satisfy(46)PuC≤r≥NωNχr.We will need a lemma.Lemma 12.
Suppose thatf:(0,1)→(0,∞) is continuous and decreasing and ∫01f(t)dt<∞ then limr→0rf(r)=0. Equivalently (setting f(t)=g(1/t)), suppose that g:(1,∞)→(0,∞) is continuous and increasing and ∫1∞g(t)t-2dt<∞ and then limt→∞g(t)/t=0.Proof.
Suppose instead thatlimsupt→∞g(t)/t>0 and let then ε>0 and tj>0 be an increasing sequence such that tj→j∞ and for all j we have g(tj)>εtj: then (47)1ε∫1∞gtt-2dt≥∑j=1∞∫tjtj+1tjt-2dt=∑j=1∞tj1tj-1tj+1=∑j=1∞tj+1-tjtj+1=∞ where the last step is explained in exercise 11 in Section 3 in [9].The above example can be finetuned as follows.Proposition 13.
Letσ:[1,∞)→[1,∞) be a continuous increasing function such that σ(1)=1, limt→∞σt=∞, and(48)∫01σ1tdt=∫1∞σtt-2dt<∞,and we will build a compact set, following the previous example, such that for r<1/2(49)PuC≤r≥NωNσ1r-1-1.Proof.
Letτ:[1,∞)→[1,∞) be the inverse τ=σ-1, and then τ(1)=1 and limt→∞τ(t)/t=∞. The subgraph(50)s,t:1<s<∞,0<t<1τs coincides with the subgraph(51)s,t:0<t<1,1<s<σ1t so(52)∫1∞1τsds=∫01σ1tdt<∞.
Letmh=⌈τ(h)⌉; note that m1=1. Since 1/τ(t)≥1/mh for h≥2 and t∈[h-1,h] so ∑h=1∞1/mh<∞; so we can build C as in the previous examples.
WhenN≥2 we let rh=4/(4mh-2) as before; note that rhmh≥1. We define χ as in (43) and note that k=χ(r) is characterized by(53)rk+1≤r<rk. Since τ(h)+1≥mh≥τ(h) then(54)τk+1+1≥mk+1≥1rk+1≥1r and hence for r<1/2(55)χr≥σ1r-1-1.Combining this with relation (46) we obtain (49).
WhenN=1 then we just set rh=1/mh and proceed similarly.


### 4.1. Proof

We eventually prove Theorem3.Proof.
Possibly adding a constant toψ and large disks to C, we assume that ψ(1/2)=NωN-1. We relate(56)t=1r-1,r=1t+1 for t≥1 and 0<r≤1/2; we let(57)σt=1+ψ1/t+1NωN and then σ:[1,∞)→[1,∞) is continuous and increasing, σ(1)=1, limt→∞σ(t)=∞, and(58)∫01σ1tdt=∫011+ψt/t+1NωNdt=1NωN+1NωN∫01/2ψs1s-12ds<∞so (48) is satisfied. By Lemma 12limr→0rψ(r)=0. We then build C using the previous examples; by (55) and (46)(59)PuC≤r≥NωNχr≥NωNσ1r-1-1=ψr.


## 4.1. Proof

We eventually prove Theorem3.Proof.
Possibly adding a constant toψ and large disks to C, we assume that ψ(1/2)=NωN-1. We relate(56)t=1r-1,r=1t+1 for t≥1 and 0<r≤1/2; we let(57)σt=1+ψ1/t+1NωN and then σ:[1,∞)→[1,∞) is continuous and increasing, σ(1)=1, limt→∞σ(t)=∞, and(58)∫01σ1tdt=∫011+ψt/t+1NωNdt=1NωN+1NωN∫01/2ψs1s-12ds<∞so (48) is satisfied. By Lemma 12limr→0rψ(r)=0. We then build C using the previous examples; by (55) and (46)(59)PuC≤r≥NωNχr≥NωNσ1r-1-1=ψr.


## 5. Alternative Proof

We now sketch a different proof of Theorem1.


### 5.1. Standard Results

We will need these standard results in Geometric Measure Theory. In the following1E will be the characteristic function of a set E.Proposition 14 (Fleming–Rishel coarea formula [10]).
LetΩ⊆RN be open and f:Ω→R be locally integrable, and then(60)DfΩ=∫RPf≤t,Ωdt where(61)DfΩ=defsup∫Ωfdiv⁡gdx:g∈Cc∞Ω→Rn,gx≤1∀x∈RN is the total mass of the distribution Df and (62)PE,Ω=defD1EΩis the perimeter of a Borel set E inside Ω.(For a proof, see also Theorem 3.40 in [11]). We will write P(E) for P(E,RN).Proposition 15 (Federer coarea formula [3]).
LetΩ⊆RN be a Borel set and f,g:Ω→R with f Lipschitzian and g integrable, and then(63)∫Ωgx∇fxdx=∫R∫f=t∩Ω)gydHN-1ydt. In particular(64)∫Ω∇fxdx=∫RHN-1f=t∩Ωdt.Proposition 16.
LetΩ⊆RN be an open set; let E be Borel and P(E,Ω)<∞. Let ∂∗E be thereduced boundary of E and ν be the inward normal, and then(65)D1E=νD1E.For definitions and proofs see, e.g., those in Section 3.5 in [11] or in Section 3 in [8].Proposition 17.
LetΩ⊆RN be an open set; let E be Borel and P(E,Ω)<∞. Let ∂∗E be thereduced boundary of E, and then (i)
∂∗E¯=∂E;(ii)
(66)PE,A=∫AD1E=Hn-1A∩∂∗Efor anyA⊆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.Proposition 18 (structure theorem).
IfE is a Borel set such that P(E)<∞ then(67)∫E∂xigxdx=-∫∂∗EgxνixdHn-1x for all g∈C01(RN) and i=1,…,n, where ∂∗E is the reduced boundary and ν(x) is the inward normal. (For a definition, see also Theorem 3.54 in [11]).


### 5.2. Lemmas

Lemma 19.
LetΩ be open; suppose that f:Ω→R is Lipschitz; let A⊆Ω open. For almost all r>0 we have that(68)Pf≤r,A=HN-1∂∗f=t∩A=HN-1f=t∩A.The proof follows comparing the Fleming–Rishel and the Federer coarea formula.Lemma 20.
LetΦ:RN→[0,∞) be an integrable compactly supported function. Let ϕ:RN→R be convex. Then the convolution Φ∗ϕ is convex.This is easily proved, e.g., by showing that the derivatives oft∈R↦(Φ∗ϕ)(x+tv) are monotone nondecreasing.LetC be compact. It is well known that uC2 is semiconcave; we provide a simple quantitative proof.Lemma 21.
u C 2 ( x ) - | x | 2 is concave.Proof.
Indeed(69)φx=uC2x-x2=miny∈C⁡x-y2-x2=miny∈C⁡x-y2-x2=miny∈C⁡-2x·y+y2 but(70)x↦-2x·y+y2 is affine hence φ is concave.Lemma 22.
Let againψ(x)=uC2(x); it is well known that (i)
∇uC(x) and ψ(x) are locally Lipschitz (by direct proof)(ii)
hence for almost anyx the differentials exist,(iii)
when these functions are differentiable andx∉C then
(a)
|∇uC(x)|=1 and |∇ψ(x)|=2ψ(x),(b)
the pointy of C closest to x is unique and is y=x-∇ψ(x)/2;
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 Theorem1.Proof.
In the first part of this proof we assume thatP(C)<∞.
LetFt={uC≤t} and Et={uC=t} in the following. The idea of the proof is as follows. Set ψ(x)=uC2(x). Let K={0<uC≤r}=Fr∖C, supposing that the boundary of K is smooth (in this case ∂Ft=Et.), and we note that ∂K=∂C∪∂Fr; we have that ∇ψ(x)=0 for x∈∂C whereas ∇ψ(x)=-2uC(x)ν=-2rν(x) for x∈∂Fr, where ν(x) is the inward normal to ∂Fr; so we use the Gauss–Green formula and write (71)2NLK≥∫KΔψxdx=-∫Et∑i=1n∂xiψxνixdHN-1x=2rHn-1Et, where the first inequality follows from Lemma 21 that implies that Δ(uC2)≤2N in the sense of distributions.
We now provide the general proof.
We sketch the following facts:(1)
For almost allt>0 we have that P(Ft)<∞, by the coarea formula. (Note that, by the proof in Section 3, this is actually truefor every  t>0.)(2)
∂Ft⊆Et by Proposition 9, so ∂∗Ft⊆Et for any t>0 such that P(Ft)<∞.(3)
By Lemma19 for almost any r>0 we have Hn-1(Et∖∂∗Ft)=0.(4)
Hence for almost anyr>0 and Hn-1–almost any x∈Er we have that(72)∇uCx=∇ψx2ψx=-νxthat is the outward normal. This follows from the existence of a “weak” tangent space of∂∗Fr near x; see Theorem 3.8 in [8].(5)
LetΦh:RN→[0,∞) a family of C∞ mollifiers, with support in D1/h and ∫RNΦh(x)dx=1. Let ψh=defψ∗Φh: then (possibly passing to a subsequence) for almost all t>0 we have(73)limh→∞⁡∫∂∗Ft∇ψh-∇ψdHn-1x=0.Indeed we know thatlimh→∞∇ψh=∇ψ in L1(FR) (for any R large), and we use the coarea formula(74)limh→∞⁡∫0Rghtdt=0,ght=∫∂∗Ft∇ψh-∇ψdHn-1x)and then (up to a subsequence)gh(t)→h0 for almost all t>0.
Letr>0 be such that all above properties are satisfied. Let K={0<uC≤r}=Fr∖C. Note that P(K)<∞ (since the boundaries of C and Fr are separated); and the essential boundary of K is the union of the essential boundaries of Fr and of C. For any point y in the topological boundary of C, limx→y∇ψ(x)=0, and then(75)∫∂∗C∇ψhxdHn-1x→h0.
By Proposition18(76)∫KΔψhxdx=∑i=1n∫K∂xixiψhxdx=-∫∂∗K∑i=1n∂xiψhxνixdHn-1x;the right hand side passes to the limit (by (75) and by (73)) (77)limh→∞⁡-∫∂∗K∑i=1n∂xiψhxνixdHn-1x=-∫∂∗Fr∑i=1n∂xiψxνixdHn-1x=∫∂∗Fr2rdHn-1x=2rHn-1∂∗Fr=2rPFr. For the left hand side of (76) we proceed as follows. By Lemma 21, φ(x)=ψ(x)-|x|2 is concave, and let(78)φh=Φh∗φ=ψh-Φh∗x2; by Lemma 20φh is concave and smooth so ∇φh≤0(79)0≥∫KΔφhdx=∫KΔψhdx-∫KΔΦh∗x2dx; then(80)limh→∞⁡∫KΔΦh∗x2dx=2Nlimh→∞⁡∫RNΦH∗1Kdx=2NLK so(81)limsuph→∞⁡∫KΔψhdx≤2NLK.
This ends the first part of the proof. Assuming thatP(C)<∞, we have proved the required result (2); namely,(82)rPuC≤r≤NL0<uC≤r, for almost any r>0, i.e., for all r>0,r∉N where N⊆R is a negligible set. Let then r>0,r∈N, and then there is a sequence rh↘hr with rh∉N; by Proposition 9 (point (2)) and the lower semicontinuity of the perimeter we deduce that (2) holds for r as well.
We eventually extend (2) to the general case. Suppose C is compact and P(C)=∞ and then there is a sequence Ch of compact sets with smooth boundary such that dH(Ch,C)→h0. Indeed we may define Ch={x:Φh∗1K(x)≥τ} where τ 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 notesMinimizing Movements held in Padova, June 20-24, 1994) of this result by Almgren, Taylor, and Wang [14].Theorem 23.
LetC⊂RN be a compact set; suppose that there are θ>0,τ>0 such that(83)Hn-1C∩Brx≥θρn-1∀x∈C,ρ∈0,τ.Then for all R>τ(84)ess-sup0<r<R⁡HN-1uC=r≤αRN-1HN-1Cwhere α=γθ-1τ1-N and γ>0 is an appropriate positive constant dependent only on the dimension N.


## 5.1. Standard Results

We will need these standard results in Geometric Measure Theory. In the following1E will be the characteristic function of a set E.Proposition 14 (Fleming–Rishel coarea formula [10]).
LetΩ⊆RN be open and f:Ω→R be locally integrable, and then(60)DfΩ=∫RPf≤t,Ωdt where(61)DfΩ=defsup∫Ωfdiv⁡gdx:g∈Cc∞Ω→Rn,gx≤1∀x∈RN is the total mass of the distribution Df and (62)PE,Ω=defD1EΩis the perimeter of a Borel set E inside Ω.(For a proof, see also Theorem 3.40 in [11]). We will write P(E) for P(E,RN).Proposition 15 (Federer coarea formula [3]).
LetΩ⊆RN be a Borel set and f,g:Ω→R with f Lipschitzian and g integrable, and then(63)∫Ωgx∇fxdx=∫R∫f=t∩Ω)gydHN-1ydt. In particular(64)∫Ω∇fxdx=∫RHN-1f=t∩Ωdt.Proposition 16.
LetΩ⊆RN be an open set; let E be Borel and P(E,Ω)<∞. Let ∂∗E be thereduced boundary of E and ν be the inward normal, and then(65)D1E=νD1E.For definitions and proofs see, e.g., those in Section 3.5 in [11] or in Section 3 in [8].Proposition 17.
LetΩ⊆RN be an open set; let E be Borel and P(E,Ω)<∞. Let ∂∗E be thereduced boundary of E, and then (i)
∂∗E¯=∂E;(ii)
(66)PE,A=∫AD1E=Hn-1A∩∂∗Efor anyA⊆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.Proposition 18 (structure theorem).
IfE is a Borel set such that P(E)<∞ then(67)∫E∂xigxdx=-∫∂∗EgxνixdHn-1x for all g∈C01(RN) and i=1,…,n, where ∂∗E is the reduced boundary and ν(x) is the inward normal. (For a definition, see also Theorem 3.54 in [11]).


## 5.2. Lemmas

Lemma 19.
LetΩ be open; suppose that f:Ω→R is Lipschitz; let A⊆Ω open. For almost all r>0 we have that(68)Pf≤r,A=HN-1∂∗f=t∩A=HN-1f=t∩A.The proof follows comparing the Fleming–Rishel and the Federer coarea formula.Lemma 20.
LetΦ:RN→[0,∞) be an integrable compactly supported function. Let ϕ:RN→R be convex. Then the convolution Φ∗ϕ is convex.This is easily proved, e.g., by showing that the derivatives oft∈R↦(Φ∗ϕ)(x+tv) are monotone nondecreasing.LetC be compact. It is well known that uC2 is semiconcave; we provide a simple quantitative proof.Lemma 21.
u C 2 ( x ) - | x | 2 is concave.Proof.
Indeed(69)φx=uC2x-x2=miny∈C⁡x-y2-x2=miny∈C⁡x-y2-x2=miny∈C⁡-2x·y+y2 but(70)x↦-2x·y+y2 is affine hence φ is concave.Lemma 22.
Let againψ(x)=uC2(x); it is well known that (i)
∇uC(x) and ψ(x) are locally Lipschitz (by direct proof)(ii)
hence for almost anyx the differentials exist,(iii)
when these functions are differentiable andx∉C then
(a)
|∇uC(x)|=1 and |∇ψ(x)|=2ψ(x),(b)
the pointy of C closest to x is unique and is y=x-∇ψ(x)/2;
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 Theorem1.Proof.
In the first part of this proof we assume thatP(C)<∞.
LetFt={uC≤t} and Et={uC=t} in the following. The idea of the proof is as follows. Set ψ(x)=uC2(x). Let K={0<uC≤r}=Fr∖C, supposing that the boundary of K is smooth (in this case ∂Ft=Et.), and we note that ∂K=∂C∪∂Fr; we have that ∇ψ(x)=0 for x∈∂C whereas ∇ψ(x)=-2uC(x)ν=-2rν(x) for x∈∂Fr, where ν(x) is the inward normal to ∂Fr; so we use the Gauss–Green formula and write (71)2NLK≥∫KΔψxdx=-∫Et∑i=1n∂xiψxνixdHN-1x=2rHn-1Et, where the first inequality follows from Lemma 21 that implies that Δ(uC2)≤2N in the sense of distributions.
We now provide the general proof.
We sketch the following facts:(1)
For almost allt>0 we have that P(Ft)<∞, by the coarea formula. (Note that, by the proof in Section 3, this is actually truefor every  t>0.)(2)
∂Ft⊆Et by Proposition 9, so ∂∗Ft⊆Et for any t>0 such that P(Ft)<∞.(3)
By Lemma19 for almost any r>0 we have Hn-1(Et∖∂∗Ft)=0.(4)
Hence for almost anyr>0 and Hn-1–almost any x∈Er we have that(72)∇uCx=∇ψx2ψx=-νxthat is the outward normal. This follows from the existence of a “weak” tangent space of∂∗Fr near x; see Theorem 3.8 in [8].(5)
LetΦh:RN→[0,∞) a family of C∞ mollifiers, with support in D1/h and ∫RNΦh(x)dx=1. Let ψh=defψ∗Φh: then (possibly passing to a subsequence) for almost all t>0 we have(73)limh→∞⁡∫∂∗Ft∇ψh-∇ψdHn-1x=0.Indeed we know thatlimh→∞∇ψh=∇ψ in L1(FR) (for any R large), and we use the coarea formula(74)limh→∞⁡∫0Rghtdt=0,ght=∫∂∗Ft∇ψh-∇ψdHn-1x)and then (up to a subsequence)gh(t)→h0 for almost all t>0.
Letr>0 be such that all above properties are satisfied. Let K={0<uC≤r}=Fr∖C. Note that P(K)<∞ (since the boundaries of C and Fr are separated); and the essential boundary of K is the union of the essential boundaries of Fr and of C. For any point y in the topological boundary of C, limx→y∇ψ(x)=0, and then(75)∫∂∗C∇ψhxdHn-1x→h0.
By Proposition18(76)∫KΔψhxdx=∑i=1n∫K∂xixiψhxdx=-∫∂∗K∑i=1n∂xiψhxνixdHn-1x;the right hand side passes to the limit (by (75) and by (73)) (77)limh→∞⁡-∫∂∗K∑i=1n∂xiψhxνixdHn-1x=-∫∂∗Fr∑i=1n∂xiψxνixdHn-1x=∫∂∗Fr2rdHn-1x=2rHn-1∂∗Fr=2rPFr. For the left hand side of (76) we proceed as follows. By Lemma 21, φ(x)=ψ(x)-|x|2 is concave, and let(78)φh=Φh∗φ=ψh-Φh∗x2; by Lemma 20φh is concave and smooth so ∇φh≤0(79)0≥∫KΔφhdx=∫KΔψhdx-∫KΔΦh∗x2dx; then(80)limh→∞⁡∫KΔΦh∗x2dx=2Nlimh→∞⁡∫RNΦH∗1Kdx=2NLK so(81)limsuph→∞⁡∫KΔψhdx≤2NLK.
This ends the first part of the proof. Assuming thatP(C)<∞, we have proved the required result (2); namely,(82)rPuC≤r≤NL0<uC≤r, for almost any r>0, i.e., for all r>0,r∉N where N⊆R is a negligible set. Let then r>0,r∈N, and then there is a sequence rh↘hr with rh∉N; by Proposition 9 (point (2)) and the lower semicontinuity of the perimeter we deduce that (2) holds for r as well.
We eventually extend (2) to the general case. Suppose C is compact and P(C)=∞ and then there is a sequence Ch of compact sets with smooth boundary such that dH(Ch,C)→h0. Indeed we may define Ch={x:Φh∗1K(x)≥τ} where τ 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 notesMinimizing Movements held in Padova, June 20-24, 1994) of this result by Almgren, Taylor, and Wang [14].Theorem 23.
LetC⊂RN be a compact set; suppose that there are θ>0,τ>0 such that(83)Hn-1C∩Brx≥θρn-1∀x∈C,ρ∈0,τ.Then for all R>τ(84)ess-sup0<r<R⁡HN-1uC=r≤αRN-1HN-1Cwhere α=γθ-1τ1-N and γ>0 is an appropriate positive constant dependent only on the dimension N.


## 6. Banach-Like Metric Spaces of Compact Sets

LetM be the family of all nonempty compact subsets of RN. In 2015 Duci and Mennucci [1] studied a family of distances dp,φ on M, where p∈[1,∞) ([1] also includes the case p=∞, which will not be considered here) and φ:[0,∞)→(0,∞) is a fixed given function. The distance was then defined, for A,B∈M, as(85)dp,φA,B=defφ∘uA-φ∘uBLpRN.Lemma 24 (Lemma 6.2 in [1]).
Suppose thatφ is monotonically strictly decreasing and(86)φx∈LpRN.Let C⊆RN be closed and nonempty; then the following are equivalent: (a)
φ∘uC∈Lp(RN).(b)
C is bounded (and then C is compact).So, for anyφ satisfying the above hypotheses, the formula in (85) would properly define a distance on M.


### 6.1. Further Results on(M,dp,φ)

Assuming appropriate hypotheses onφ, [1] proved many interesting properties of the metric space (M,dp,φ).Hypotheses 25 (Hypotheses 6.1 in [1]).
Suppose thatφ is monotonically strictly decreasing and of class C1, such that(87)φx∈LpRN.The resulting metric spaces enjoy many interesting properties.(i)
dp,φ is Euclidean invariant.(ii)
The topology induced bydp,φ over the space M coincides with the topology induced by the Hausdorff distance (Theorem 6.11 in [1]).(iii)
The metric space(M,dp,φ) is complete (Theorem 6.12 in [1]).


### 6.2. Induced Distance and Geodesics

Given a metric space(M,d), we define thelengthLendγ of a continuous path γ:[α,β]→M by using thetotal variation(88)Lendγ=defsupT⁡∑i=1ndγti-1,γti,where the supremum is computed over all finite subsets T={t0,…,tn} of [α,β] and t0≤⋯≤tn. We then define theinduced distancedg by(89)dgx,y=definfγ⁡Lendγ,where the infimum is taken in the class of all continuous paths γ connecting x to y. If the infimum is a minimum, the path providing the minimum is called a geodesic.With further hypotheses onφ(t), more can be said on the metric space (M,dp,φ).Hypotheses 26 (Hypotheses 6.17 in [1]).
Letφ be as defined in Hypotheses 25. We moreover suppose that there is a constant T>0 such that φ(t) is convex for t≥T, and(90)φ′x∈LpRN.Then these results follow.(i)
The space(M,dp,φ) is Lipschitz–arc connected (Prop. 6.18 in [1]).(ii)
For anyρ>0,A∈M,(91)DgA,ρ=defB∈M∣dp,φgA,B≤ρis compact in the(M,d) topology (where dp,φg is the distance induced by dp,φ) (Theorem 6.20 in [1]).(iii)
Hence any two compact sets are connected by a geodesic in(M,dp,φ).(iv)
Whenp∈(0,∞) 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)
Forp=2 there is an interpretation of the metric dp,φg 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 Hypotheses25 and Hypotheses 26 can be satisfied by simple functions such as φ(t)=exp⁡(-t) or φ(t)=(1+t)-α for α>N/p.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 Lemma24 it was assumed that φ bemonotonically strictly decreasing. This was a useful ingredient in writing a simple direct proof of the implication b⇒a. Some geometrical and analytical considerations though suggested that this hypothesis was not strictly needed.Indeed as a consequence of Theorem1 we can now prove Theorem 4.Proof.
By Lemma19 and Theorem 1 for almost any r>0,(92)rHN-1uC=r≤NLNuC≤r.
Let thenr¯>0 be such that C⊆Dr¯(0), and then {uC=r}⊆D(r+r¯); so for almost all r>0(93)HN-1uC=r≤Nrr+r¯N.
Letφ:[0,∞)→[0,∞) be as in hypotheses; we know (Lemma 22) that |∇uC(x)|=1 for almost all x∈RN∖C, and then by the Federer coarea formula Proposition 15(94)∫uCx>εφuCxpdx=∫uCx>εφuCxp∇uCxdx=∫ε∞∫uC=t∩Ω)φuCxpdHN-1ydt=∫ε∞φtpHN-1uC=tdt≤∫ε∞φtpNtt+r¯Ndt<∞ by relation (6), whereas(95)∫uCx≤εφuCxpdx<∞ by (7); so φ∘uC∈Lp(RN).


## 6.1. Further Results on(M,dp,φ)

Assuming appropriate hypotheses onφ, [1] proved many interesting properties of the metric space (M,dp,φ).Hypotheses 25 (Hypotheses 6.1 in [1]).
Suppose thatφ is monotonically strictly decreasing and of class C1, such that(87)φx∈LpRN.The resulting metric spaces enjoy many interesting properties.(i)
dp,φ is Euclidean invariant.(ii)
The topology induced bydp,φ over the space M coincides with the topology induced by the Hausdorff distance (Theorem 6.11 in [1]).(iii)
The metric space(M,dp,φ) is complete (Theorem 6.12 in [1]).


## 6.2. Induced Distance and Geodesics

Given a metric space(M,d), we define thelengthLendγ of a continuous path γ:[α,β]→M by using thetotal variation(88)Lendγ=defsupT⁡∑i=1ndγti-1,γti,where the supremum is computed over all finite subsets T={t0,…,tn} of [α,β] and t0≤⋯≤tn. We then define theinduced distancedg by(89)dgx,y=definfγ⁡Lendγ,where the infimum is taken in the class of all continuous paths γ connecting x to y. If the infimum is a minimum, the path providing the minimum is called a geodesic.With further hypotheses onφ(t), more can be said on the metric space (M,dp,φ).Hypotheses 26 (Hypotheses 6.17 in [1]).
Letφ be as defined in Hypotheses 25. We moreover suppose that there is a constant T>0 such that φ(t) is convex for t≥T, and(90)φ′x∈LpRN.Then these results follow.(i)
The space(M,dp,φ) is Lipschitz–arc connected (Prop. 6.18 in [1]).(ii)
For anyρ>0,A∈M,(91)DgA,ρ=defB∈M∣dp,φgA,B≤ρis compact in the(M,d) topology (where dp,φg is the distance induced by dp,φ) (Theorem 6.20 in [1]).(iii)
Hence any two compact sets are connected by a geodesic in(M,dp,φ).(iv)
Whenp∈(0,∞) 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)
Forp=2 there is an interpretation of the metric dp,φg 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 Hypotheses25 and Hypotheses 26 can be satisfied by simple functions such as φ(t)=exp⁡(-t) or φ(t)=(1+t)-α for α>N/p.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 Lemma24 it was assumed that φ bemonotonically strictly decreasing. This was a useful ingredient in writing a simple direct proof of the implication b⇒a. Some geometrical and analytical considerations though suggested that this hypothesis was not strictly needed.Indeed as a consequence of Theorem1 we can now prove Theorem 4.Proof.
By Lemma19 and Theorem 1 for almost any r>0,(92)rHN-1uC=r≤NLNuC≤r.
Let thenr¯>0 be such that C⊆Dr¯(0), and then {uC=r}⊆D(r+r¯); so for almost all r>0(93)HN-1uC=r≤Nrr+r¯N.
Letφ:[0,∞)→[0,∞) be as in hypotheses; we know (Lemma 22) that |∇uC(x)|=1 for almost all x∈RN∖C, and then by the Federer coarea formula Proposition 15(94)∫uCx>εφuCxpdx=∫uCx>εφuCxp∇uCxdx=∫ε∞∫uC=t∩Ω)φuCxpdHN-1ydt=∫ε∞φtpHN-1uC=tdt≤∫ε∞φtpNtt+r¯Ndt<∞ by relation (6), whereas(95)∫uCx≤εφuCxpdx<∞ by (7); so φ∘uC∈Lp(RN).



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*Source: 8283496-2019-04-24.xml*