Rigidity for the perimeter inequality under Schwarz symmetrisation

In this paper, we give necessary and sufficient conditions for the rigidity of perimeter inequality under Schwarz symmetrisation. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in \cite{fusco2013stability}, are also necessary.


Introduction
Symmetrisation procedures have a wide range of applications in modern analysis, geometric variational problems and optimisation.Understanding the behaviour of functional and perimeter inequalities under symmetrisation allows to prove the existence of symmetric minimisers of geometric variational problems, and to provide comparison principles for solutions of PDEs (see, for instance [15,16,24,25] and the references therein).
Examples of set symmetrisations under which the volume is preserved and the perimeter does not increase include Steiner symmetrisation, Ehrhard symmetrisation, circular and spherical symmetrisation.We say that rigidity holds for a perimeter inequality if the set of extremals is trivial.Showing rigidity can lead to proving the uniqueness of minimisers of variational problems.For example, proving the rigidity of Steiner's inequality for convex sets was substantial in the celebrated proof of the Euclidean isoperimetric inequality by De Giorgi (see, [10,11]).
Later on, the study of rigidity was revived in the seminal paper of Chlebík, Cianchi and Fusco (see [8]), where the authors gave the sufficient conditions for rigidity of Steiner's inequality, also for sets that are not convex.Henceforth, necessary and sufficient conditions for rigidity for Steiner's inequality have been obtained in [5] in the case where the distribution function is a Special function of Bounded Variation with locally finite jump set.In the Gauss space, necessary and sufficient conditions for rigidity of Ehrhard's inequality are given in [6].In the last two papers, the results are stated in terms of essential connectedness.For an expository article of the aforementioned rigidity results, we refer to [4].In [7], the authors provided the necessary and sufficient conditions for rigidity for perimeter inequality under spherical symmetrisation, while in [20], sufficient conditions for rigidity have been given for the anisotropic Steiner's perimeter inequality.We further point out that, regarding the smooth case, the authors in [18] proved sufficient conditions for rigidity of perimeter inequality in warped products, for a wide class of symmetrisations, including Steiner, Schwarz and spherical symmetrisation.
The literature about Steiner's perimeter inequality of a higher codimension is less explored.Particularly, sufficient conditions for rigidity for any codimension have been provided in [2], through a comprehensive analysis of the barycenter function.The problem of a complete characterisation (that is, necessary and sufficient conditions) for the rigidity of generic higher codimensions, however, remains open.
A special case of interest is where the codimension is equal to (n − 1).In this case, Steiner's symmetrisation of codimension (n − 1) is usually referred to as Schwarz symmetrisation.
The purpose of this paper is to provide necessary and sufficient conditions for rigidity of equality cases for the perimeter inequality under Schwarz symmetrisation.In particular, we prove that the sufficient conditions for rigidity shown in [2] are also necessary.Our results are established by following techniques developed in [7].
In the remainder of this introductory section, we recall the setting of the problem, and we state our main results.
1.1.Schwarz symmetrisation.For n ≥ 2 with n ∈ N, we label each point x ∈ R n as x = (z, w), where z ∈ R and w ∈ R n−1 .
Given a measurable set E ⊂ R n and z ∈ R, we define the (n − 1)-dimensional slice of E at z as For a Lebesgue measurable function ℓ : R → [0, ∞), we say that the set where We can associate to ℓ the function r ℓ : R → [0, ∞), which is such that where B n−1 (w, ρ) denotes the open ball in R n−1 with radius ρ and centered at w ∈ R n−1 .Note that r ℓ (z) is the radius of an (n − 1)-dimensional ball whose measure is ℓ(z), and can be explicitly written as where we set ω n−1 := H n−1 (B n−1 (0, 1)).If E ⊂ R n is ℓ-distributed, then the Schwarz symmetric set F ℓ of E with respect to the axis {w = 0} is defined as (1.4) see Figure 1.1.This is the ℓ-distributed set whose cross sections are (n − 1)-dimensional open balls centred at the z axis.We notice that the Schwarz symmetric set F ℓ of an ℓ-distributed set E depends only on the function ℓ, and not on the particular ℓ-distributed set E under consideration.
Due to Fubini's theorem, Schwarz symmetrisation preserves the volume, i.e. if E is ℓdistributed and H n (E) < ∞, it turns out that H n (E) = H n (F ℓ ).Moreover, the perimeter inequality under Schwarz symmetrisation holds, that is (1.5) Here, P (E) stands for the perimeter of E in R n (see Section 2.4).The inequality (1.5) is well-known in the literature (see, for instance, [3], where this is proved through a careful approximation by polarisations).In [2], one can find an alternative and direct proof, which allowed the authors to give sufficient conditions for rigidity of Steiner's inequality of a general higher codimension k, where 1 in case of n = 3.Note that, in general the slices of the set E do not need to be disks.1.2.Rigidity for perimeter inequality under Schwarz symmetrisation.We shall now describe the main objective of the present paper.Given a Lebesgue measurable function ℓ : R → [0, ∞), such that F ℓ is a set of finite perimeter and finite volume, we define the class of equality cases of (1.5) as ( Due to the invariance of the perimeter under translations along a direction τ ∈ R n−1 , as well as the definition of the symmetric set F ℓ , the following inclusion is always true: where △ denotes the symmetric difference of sets.We say that rigidity holds for (1.5) if the opposite inclusion is also satisfied, i.e.
1.3.State of the art.Let us now give an account of the available results in the literature for the rigidity of (1.5).In general, not all equality cases of (1.5) can be written as a translation of the symmetric set F ℓ .This can happen, for instance, if the (reduced) boundary ∂ * F ℓ of F ℓ contains flat vertical parts.In such a case, we can find an ℓ-distributed set E which preserves perimeter under symmetrisation, and it is not equivalent to (a translation of) the symmetric set F ℓ ; see Figure 1.2.
In order to rule out this issue, the authors in [2] localised the problem, by considering an open set Ω ⊂ R, and imposing the following condition: where ν F ℓ w (z, w) denotes the w-component of the measure-theoretic outer unit normal to the symmetric set F ℓ .It turns out that (1.8) is related to the regularity of the function ℓ.Note that, in general, if E is a set of finite perimeter in R n , then either F ℓ is equivalent to R n , or ℓ is a function of Bounded Variation in R (see Proposition 2.2).
In [2, Proposition 3.5], the authors showed that (1.8) is equivalent to asking that ℓ is a Sobolev function in Ω, as explained below.(1.8).Note that the function ℓ is discontinuous at z, so that also (1.9) is violated.
Proposition 1.1.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume in R n and let Ω ⊂ R be an open set.Then Even if condition (1.8) (or, equivalently, (1.9)) is satisfied, rigidity can still be violated.In particular, this can happen when the symmetric set F ℓ is not connected in a suitable measure-theoretic way, despite the fact that it can be connected from a topological point of view; see Figure 1   Note that, once condition (1.8) (or, equivalently, (1.9)) is imposed, we have that ℓ ∈ W 1,1 (Ω), and since Ω is a one-dimensional set, ℓ is absolutely continuous in Ω. Therefore the condition imposed in [2] to rule out situations as in Figure 1.3 can be written as Here, P (E; Ω × R) denotes the relative perimeter of E in Ω × R.
1.4.The main result.Our contribution is to show that conditions (1.8) and (1.10) are also necessary for rigidity.As we have already observed, the proof of the Theorem 1.2 requires the localisation of the problem in an open and connected set Ω ⊂ R, to impose the condition (1.9).We will show that this can be avoided.We also notice that, if F ℓ is a set of finite perimeter and finite volume, in general, we only have that ℓ ∈ BV (R) and this means that ℓ may be discontinuous.Therefore, we need to rephrase condition (1.10) in terms of the approximate lim inf ℓ ∧ of ℓ at every point z ∈ R, see Section 2. We are now able to state our main result.Below, J denotes the interior of J.
Theorem 1.3.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume.Then, the following statements are equivalent: As we have already pointed out, the proof of the direction (ii) =⇒ (i) of Theorem 1.3 relies on the proof of [2, Theorem 1.2].We will prove that the converse (i) =⇒ (ii) is also true.We would like to emphasise that our approach does not lie on the comprehensive use of a general perimeter formula for sets E ⊂ R n satisfying equality in (1.5), as it appears in [6].On the contrary, inspired by the techniques developed in [7], we analyse the properties of the function ℓ and we provide a careful study of the transformations that can be applied on the symmetric set F ℓ , without creating any perimeter contribution.
To this end, the rest of the paper is structured as follows.In Section 2, we fix the notation, build the necessary background and we gather some preliminary results that appeared in the literature.In Section 3, we show the direction (i) =⇒ (ii) of Theorem 1.3, by studying the properties of the distribution function ℓ, and exploiting counterexamples where rigidity is violated.

Background and proof of the Theorem 1.3 (ii) =⇒ (i)
In this section, we will recall the necessary machinery, which will be used throughout the paper.The interested reader could refer to [1,2,12,13,17,23].
We fix n ∈ N, with n ≥ 2. For each x ∈ R n , we write x = (z, w), with z ∈ R and w ∈ R n−1 .The standard Euclidean norm will be denoted by | • | in R, R n−1 or R n depending on the context.For 1 ≤ m ≤ n, we will denote the m-dimensional Hausdorff measure in R n by H m .For every radius ρ > 0 and x ∈ R n we write B ρ (x) for the open ball of R n with radius ρ and centered at x.The volume of the unit ball in R n is denoted as ω n , i.e. ω n := H n (B 1 (0)).Note that throughout the paper, in case of balls in different dimensions, we will denote the corresponding ball in dimension m with radius ρ centred at w ∈ R m by writing B m (w, ρ).Now, for x ∈ R n and ν ∈ ∂B 1 (0), we set Let {E j } j∈N be a sequence of Lebesgue measurable sets in R n with H n (E j ) < ∞ for every j ∈ N, and let E ⊂ R n be a Lebesgue measurable set with H n (E) < ∞.We say that {E j } j∈N converges to E as j → ∞ and we write where △ stands for the symmetric difference of sets.Additionally, if E 1 , E 2 ⊂ R n are Lebesgue measurable sets, we say that and Moreover, the characteristic function of a Lebesgue measurable set E ⊂ R n will be denoted by χ E .
2.1.Density points.Let E ⊂ R n be a Lebesgue measurable set and x ∈ R n .We define the lower and upper n-dimensional densities of E at x as respectively.The maps x −→ θ * (E, x) and x −→ θ * (E, x) are Borel functions (even in case where E is Lebesgue non-measurable) and they coincide H n -a.e. in R n .Hence, the n-dimensional density of E at x is defined as the Borel function For each s ∈ [0, 1], we define the set of points of density s with respect to E as The essential boundary ∂ e E of E is defined as the set 2.2.Approximate limits of measurable functions.Let g : R n → R be a Lebesgue measurable function.We define the approximate upper limit g ∨ (x) and the approximate lower limit g ∧ (x) of g at x ∈ R n as respectively.We highlight the fact that both g ∨ and g ∧ are Borel functions and they are defined for every x ∈ R n with values in R ∪ {±∞}.In addition, if g 1 : R n → R and g 2 : R n → R are measurable functions such that x) for every x ∈ R n .The approximate discontinuity set S g of g is defined as and satisfies H n (S g ) = 0.Moreover, even if g ∧ , g ∨ could take values ±∞ on S g , it turns out that the difference g ∨ − g ∧ is well-defined in R ∪ {±∞} for every point x ∈ S g .In the light of the above considerations, the approximate jump [ g ] of g is the Borel function Let E ⊂ R n be a Lebesgue measurable set.We will say that s ∈ R ∪ {±∞} is the approximate limit of g at x with respect to E, denoted by s = aplim(g, E, x), if for every M > 0 (s = +∞), and θ ({g > −M } ∩ E; x) = 0, for every M > 0 (s = −∞).We will say that x ∈ S g is a jump point of g if there exist ν ∈ ∂B 1 (0) such that g ∨ (x) = aplim(g, H + x,ν , x) and g ∧ (x) = aplim(g, H − x,ν , x).In this spirit, we define the approximate jump direction ν g (x) of g at x as ν g (x) := ν.The set of approximate jump points of g is denoted by J g .Note that J g ⊂ S g and ν g : J g → ∂B 1 (0) is a Borel function.

Functions of Bounded Variation.
Let Ω ⊂ R n be an open set.We denote by C 1 c (Ω; R n ) and by C c (Ω; R n ) the class of C 1 functions with compact support and the class of all continuous functions with compact support from Ω to R n , respectively.We also recall the Sobolev space W 1,1 (Ω), that is, the space of all functions g ∈ L 1 (Ω), whose distributional derivative Dg belongs to L 1 (Ω).
Given g ∈ L 1 (Ω), the total variation of g in Ω is defined as We then define the space of functions of bounded variation in Ω, denoted by BV (Ω), as the set of functions g ∈ L 1 (Ω) such that |Dg|(Ω) < ∞.In addition, we will say that g ∈ BV loc (Ω), if g ∈ BV (Ω ′ ) for every Ω ′ ⊂⊂ Ω.If g ∈ BV (Ω), due to Radon-Nikodym decomposition of Dg with respect to H n , we have where D ac g and D s g are mutually singular measures and D ac g ≪ H n .The density of D ac g with respect to H n will be denoted as ∇g, and we have that ∇g ∈ L 1 (Ω, R n ) with D ac g = ∇g dH n .Additionally, it turns out that H n−1 (S g \J g ) = 0 and [ g ] ∈ L 1 loc (H n−1 J g ).The jump part of g is the R n -valued Radon measure given by Finally, the Cantorian part D c g of Dg is defined as the R n -valued Radon measure and is such that |D c g|(N ) = 0 for every set N ⊂ R n , which is σ-finite with respect to ) can be decomposed as the sum g = g ac + g j + g c , (2.4) where g ac ∈ W 1,1 (a, b), g j is a purely jump function (that is, Dg j = D j g j ) and g c is a purely Cantorian function (that is, Dg c = D c g c ) (see [1,Corollary 3.33]).Moreover, the total variation |Dg| of Dg can be written as where the supremum is taken over all M ∈ N and over all possible partitions of the interval (a, b) 2.4.Sets of locally finite perimeter in the Euclidean space.Let n, m ∈ N with 1 ≤ m ≤ n.Let also E ⊂ R n be an H m -measurable set.We say that E is a countably H m -rectifiable set if there exist a countable family of Lipschitz functions (g j ) j∈N , where Let E ⊂ R n be a Lebesgue measurable set.We say that E is a set of locally finite perimeter in R n if there exists an R n -valued Radon measure µ E , such that Note that, E is a set of locally finite perimeter if and only if is a Borel set, then the relative perimeter of E in G is defined as When G = R n , we ease the notation to P (E) exists and belongs to ∂B 1 (0).
The Borel function ν E : ∂ * E → ∂B 1 (0) is usually referred to as the measure-theoretic outer normal to E. Due to Lebesgue-Besicovitch derivation theorem and [1, Theorem 3.59], it holds that the reduced boundary Thus, for every Borel set G ⊂ R n we have that Finally, if E is a set of locally finite perimeter, it holds 2.5.Preliminary results.In this final subsection, we state some results which will be useful in the following.
The first significant result relates to the set E z defined in (1.1).Namely, as it turns out, for H 1 -a.e z ∈ R, E z is a set of finite perimeter and its reduced boundary ∂ * (E z ) enjoys an advantageous property.These facts follow due to a variant of a result by Vol'pert [26], which is provided in [2, Theorem 2.4].
Proposition 2.1 (Vol'pert).Let E be a set of finite perimeter in R n .Then for H 1 -a.e.z ∈ R the following hold true: Thanks to (ii) above, we will often write ∂ * E z instead of (∂ * E) z or ∂(E z ).The next result presents a crucial regularity property of the function ℓ, and it can be found in [2, Lemma 3.1].Proposition 2.2.Let E be a set of finite perimeter in R n .Then either ℓ(z) = ∞ for H 1 -a.e.z ∈ R, or ℓ(z) < ∞ for H 1 -a.e.z ∈ R and H n (E) < ∞.In the latter case, we have ℓ ∈ BV (R).
We present the following auxiliary inequality, which is a special case of [2, Proposition 3.4].
Proposition 2.3.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume.Let E ⊂ R n be an ℓ-distributed set and let f : R → [0, ∞] be a Borel measurable function.Then Moreover, if E = F ℓ , the equality holds in (2.8).
A straightforward consequence of the above result is the following.
Corollary 2.4.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume.Then for every Borel set B ⊂ R.
For sake of completeness, we close this preliminary section by presenting the proof of Theorem 1.3 (ii) =⇒ (i).
Thus, it turns out that (1.10) is true.Therefore, due to Theorem 1.2, (i) follows.

Proof of the Theorem 1.2 (i) =⇒ (ii)
We start our analysis with the following lemma, which will be extensively used in the sequel.
Lemma 3.1.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume.Let also r ℓ be defined as in (1.3) and consider z ∈ R. Then Proof.The proof is divided into two steps.
Step 2: We conclude the proof.We first observe that by Corollary 2.4 with B = {z}, we obtain Finally, recalling that, by Step 1 , which concludes the proof.Now, we can show that if the set {ℓ ∧ > 0} fails to be a (possibly unbounded) interval, then rigidity is violated.Proposition 3.2.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume, and let r ℓ be defined as in (1.3).Suppose that the set {ℓ ∧ > 0} is not an interval.That is, suppose that there exists z ∈ {ℓ ∧ = 0} such that Then, rigidity is violated.More precisely, setting Proof.Let E 1 , E 2 and E be as in the statement.Let also τ ∈ R n−1 .First of all, note that, since {z < z} is open and E ∩ {z < z} = F ℓ ∩ {z < z}, we have ℓ ∩ {z < z}. for every s ∈ [0, 1].In accordance of that, we infer In the same fashion, for every τ ∈ R n−1 , we obtain Hence, due to (3.5) and (3.6), we have As a consequence, in order to complete the proof, we need to show that We divide the proof of (3.7) in several steps.
Step 1c: To conclude the proof of Step 1, we observe that, by Step 1a and 1b, as well as by the definition of E, it follows that Therefore, Step 2: Finally, we show (3.7).Note that, thanks to Step 1, Lemma 3.1 and perimeter inequality (1.5), we have which makes our proof complete.
We will now show that, if the jump part D j ℓ of Dℓ is non-zero, then rigidity is violated.Proposition 3.3.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume, and let r ℓ be defined as in (1.3).Suppose that ℓ has a jump at some point z ∈ R. Then rigidity is violated.More precisely, setting Proof.Let E 1 , E 2 and E be as in the statement.Let also τ ∈ R n−1 be such that (3.12) is satisfied.It is not restrictive to assume that By an analogous argument as in the beginning of the proof of Proposition 3.2, we obtain Hence, in order to complete the proof, we finally need to show that We divide the proof of (3.14) into further steps.
Step 1: We prove that In order to show (3.15), it suffices to prove that (3.16b) First, let us prove (3.16a).To achieve that, we observe, due to (2.6), our claim will follow if we prove that To this end, suppose that w ∈ R n−1 is such that |w − τ | < r ∧ ℓ (z).Then, we observe that Hence, to complete the proof of the claim, it remains to show that lim Then there exists δ > 0 such that Then, employing (3.20) and the definition of the set E, we infer Moreover, we note that for ρ ∈ (0, ρ), we have As a consequence, for ρ ∈ (0, ρ) we obtain Then, thanks to (3.13), we infer Making use of similar arguments as above, it can be shown that which, shows (3.16b), and in turn (3.15).
Step 2: We conclude the proof.From (3.12), we infer that ).As a consequence, thanks to Step 1, Lemma 3.1 and perimeter inequality (1.5), we have From this, we deduce (3.14), which completes the proof.
We are going to prove now that if the Cantorian part D c ℓ of Dℓ is non-zero, then rigidity is violated.Proposition 3.4.Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume.Let also r ℓ be as in (1.3).Suppose that D c ℓ = 0. Then rigidity is violated.
Proof.With no loss of generality, we assume that ℓ is a purely Cantorian function.Indeed, one can decompose ℓ as ℓ = ℓ a + ℓ j + ℓ c , (3.23) where ℓ a ∈ W 1,1 (R), ℓ j is purely jump function and ℓ c is purely Cantorian.In the case of ℓ j = 0, the result becomes trivial since, due to Proposition 3.3, rigidity is violated.Furthermore, in the case of ℓ = ℓ c , thanks to (3.23), the following proof will sustain only to ℓ c .Thus, in what will follow, we assume that In addition, due to Proposition 3.2, we can assume that {ℓ ∧ > 0} is an interval, otherwise, the result becomes trivial.Note now that, since ℓ is continuous, there exists a, b > 0 such that J := (a, b) ⊂⊂ {ℓ ∧ > 0} and ℓ(z) > 0, for every z ∈ J.
We now fix λ ∈ (0, 1), and we define the function g : R → R as .
Let us fix a unit vector e ∈ R n−1 .We define the set One can observe that we cannot obtain E using a single translation on F ℓ along R n−1 .We are going to prove now that E ∈ K(ℓ).We divide the proof in several steps.
Step 1: We construct a sequence {ℓ k } k∈N , where ℓ k : J → [0, ∞), which satisfies the following properties: (i) r k ℓ (z) −→ r ℓ (z), as k → ∞ for H 1 -a.e.z ∈ J, (ii) Dℓ k = D j ℓ k for every k ∈ N, (iii) lim k→∞ P (F ℓ k ; J × R n−1 ) = P (F ℓ ; J × R n−1 ).By (2.5) and since ℓ is continuous, we have where the supremum runs over N ∈ N and over all z 1 , z 2 , . . ., z N with a < z It is not restrictive to assume that the partitions are increasing in k, i.e.
where we set z k 0 := a and z k N k +1 := b.Moreover, we set for every z ∈ J and for every k ∈ N.
Step 2: For k ∈ N, we will construct a ℓ k -distributed set E k satisfying As a consequence of (ii) in Step 1, for k ∈ N we infer that Dr k ℓ = D j r k ℓ and that the jump set of r ℓ k is a finite set.In particular, where, for each i ∈ {1, 2, . . ., N k }, δ z k i denotes the Dirac delta measure concentrated at the point z k i .Let us now fix λ ∈ (0, 1) and we define iteratively the family of sets Applying Proposition 3.2 for each i ∈ {1, . . ., N k }, we infer that Note now, that for i ∈ {1, 2, • • • , N k } the general term of the above family of sets can be written as Therefore, if we set we conclude that P (E k ; J × R n−1 ) = P (F ℓ k ; J × R n−1 ), for every k ∈ N. Step 3: We claim now, that for some ℓ-distributed set E satisfying P ( E; J × R n−1 ) = P (F ℓ ; J × R n−1 ).Indeed, thanks to (i) of Step 1, it turns out that r k ℓ (z) −→ r ℓ (z) for H 1 -a.e.z ∈ J.As a result, recalling (3.32) and if E is defined as E := (z, w) ∈ J × R n−1 : |w − λ(r ℓ (z) − r ℓ (a))e| < r ℓ (z) , (3.33) it follows that E is ℓ-distributed and Finally, by Step 2, lower semicontinuity of perimeter with respect to L 1 convergence (see e.g.[17,Theorem 12.15]) and perimeter inequality (1.5), we obtain P (F ℓ ; J × R n−1 ) ≤ P ( E; J × R n−1 ) ≤ lim inf k→∞ P (E k ; J × R n−1 ) = lim inf k→∞ P (F ℓ k ; J × R n−1 ) = lim k→∞ P (F ℓ k ; J × R n−1 ) = P (F ℓ ; J × R n−1 ), and thus, P ( E; J × R n−1 ) = P (F ℓ ; J × R n−1 ).
Therefore, E ∈ K(ℓ).In the light of this, the proof is completed.
We can now show the implication (i) =⇒ (ii) of Theorem 1.3.
Proof of Theorem 1.2: (i) =⇒ (ii).Assume that E ∈ K(ℓ).Then, in order to show the result, it is enough to combine Proposition 3.2, Proposition 3.3 and Proposition 3.4.

4 Figure 3 . 3 .
Figure 3.3.A graphical illustration of the set E k N k in Step 2.
Let ℓ : R → [0, ∞) be a measurable function, such that F ℓ is a set of finite perimeter and finite volume.Let Ω ⊂ R be a connected open set, and suppose that (1.8) and (1.10) are satisfied.If