Rigidity and large volume residues in exterior isoperimetry for convex sets

A comparison theorem by Choe, Ghomi and Ritor\'e states that the exterior isoperimetric profile $I_\mathcal{C}$ of any convex body $\mathcal{C}$ in $\mathbb{R}^N$ lies above that of any half-space $H$. We characterize convex bodies such that $I_\mathcal{C}\equiv I_H$ in terms of a notion of"maximal affine dimension at infinity'', briefly called the asymptotic dimension $d^*(\mathcal{C})$ of $\mathcal{C}$. More precisely, we show that $I_\mathcal{C}\equiv I_H$ if and only if $d^*(\mathcal{C})\ge N-1$. We also show that if $d^*(\mathcal{C})\le N-2$, then, for large volumes, $I_\mathcal{C}$ is asymptotic to the isoperimetric profile of $\mathbb{R}^N$. We then estimate, in terms of $d^*(\mathcal{C})$-dependent power laws, the order as $v\to\infty$ of the difference between $I_\mathcal{C}$ and the isoperimetric profile of $\mathbb{R}^N$.


Introduction
In this paper we consider the exterior isoperimetric problem for convex bodies C ⊂ R N .We first identify a natural notion of "maximal affine dimension at infinity", called the asymptotic dimension of C, denoted by d * (C) and typically larger than the affine dimension of the recession cone C ∞ of C. We then prove that unbounded convex bodies with d * (C) ≥ N − 1 have the same exterior isoperimetric profile of half-spaces.We also begin the study of exterior isoperimetric profiles of convex bodies with d * (C) ≤ N − 2, and introduce in this setting the notion of isoperimetric residue.We have two motivations for presenting these results.The first one is the desire of understanding the geometric information "stored" in the large volume behavior of exterior isoperimetric profiles, as done in [12] with the introduction of the isoperimetric residues of compact (non-necessarily convex) sets.The second one is providing characterizations of rigidity of equality cases in a recent comparison theorem for exterior isoperimetric profiles of convex bodies proved by Choe, Ghomi and Ritoré in [4].
Before further discussing these points we need to introduce some notation and terminology.Given a closed set C ⊂ R N , the exterior isoperimetric profile of C is the function I C : 1 (0, ∞) → (0, ∞) defined by (1.1) That is, I C is what is commonly called the isoperimetric profile of R N \ C; in particular, I ∅ (v) = N ω 1/N N v (N −1)/N coincides with the isoperimetric profile of R N .Here |E| denotes the volume (Lebesgue measure) of a Borel set E and P (E; G) its distributional perimeter, so that P (E; G) = H N −1 (G ∩ ∂E) as soon as E is open with C 1 -boundary.We also denote by H k the k-dimensional Hausdorff measure in R N and by ω N the volume of the unit ball in R N .
With this notation in hand, we can state the comparison theorem proved by Choe, Ghomi and Ritoré in [4]: if C is a convex body, that is, if C is a closed convex subset of R N with nonempty interior and different from the whole R N , then, denoting by H a generic half-space of R N , one has (1.2) To provide some context, we recall that comparison theorems of this form are one of the most studied type of results in Riemannian Geometry.For example, the Levy-Gromov comparison theorem states that among compact Riemannian manifolds (M N , g) whose Ricci tensor is bounded from below by (N − 1) g, the standard sphere (S N , g S N ) has the lowest isoperimetric profile 1 .Comparison theorems are usually accompanied by rigidity statements.Using again the Levy-Gromov comparison theorem as a model, the corresponding rigidity statement is that if the isoperimetric profiles of (M N , g) and (S N , g S N ) coincide even for just one volume fraction, then (M N , g) is isometric to (S N , g S N ).The situation with the CGR-comparison theorem is remarkably different.Indeed, as proved in [4] when C has boundary of class C 2 , and for arbitrary convex bodies in [9], one has for some value of v > 0 s.t.there exist minimizers of I C (v) (1.3) if and only if ∂C has a "facet" that supports a half-ball of volume v contained in R N \ C .(1.4) In particular, the class of convex bodies satisfying I C (v) = I H (v) for a single value of v > 0 is extremely vast, e.g., it contains all convex polyhedra!, and coincides with the class of convex bodies C such that I C = I H on an open interval.We can of course formulate the rigidity problem in a stronger sense, and consider the class of convex bodies C such that I C (v) = I H (v) for every v > 0. While it is not hard to construct unbounded convex bodies that are not half-spaces and still satisfy I C ≡ I H , there is a surprisingly direct condition that characterizes the stronger notion of rigidity I C ≡ I H .This condition can be expressed in terms of the asymptotic dimension d * (C) of C introduced right after the following statement, which is our first main result.Remark 1.2.Notice that Theorem 1.1 addresses rigidity without making the conditional assumption that I C (v) admits minimizers, either for one or more values of v.In fact, by combining Theorem 1.1 with the equivalence between (1.3) and (1.4) proved in [4,9], one easily shows that if C is strictly convex and d * (C) ≥ N − 1, then, for every v > 0, I C (v) does not admit minimizers; see Corollary 4.4.
We now introduce the notion of asymptotic dimension d * (C) of C. To begin with we recall that if x ∈ C and C is a convex body, then the family of convex sets {λ (C − x)} λ>0 is monotone increasing with respect to set inclusion.In particular, its limit as λ → 0 + , called the recession cone C ∞ of C, can be defined as The (affine) dimension dim(C ∞ ) of C ∞ (that is, the dimension of the smallest affine subspace of R N containing C ∞ ) directly quantifies the dimension of the set of directions along which C is unbounded (and, indeed, To define d * (C), and understand rigidity of the CGR-comparison theorem, we follow a related procedure.Rather than working with a fixed point x ∈ C, we consider sequences {x n } n in C and {λ n } n in (0, ∞), look at the resulting sequences of convex sets {λ n (C − x n )} n , consider all their possible accumulation points K in the Kuratowski convergence, which are still convex sets in R N , and finally maximize the affine dimension among the possible limits K.More formally and concisely, we set When d * (C) = 0, i.e., when C is bounded, this problem has been thoroughly addressed in the recent paper [12] without even assuming the convexity of C. Roughly speaking, the main result in [12] where M ranges over all the affine hyperplanes in R N and p M : R N → M denotes the orthogonal projection over M .In fact, R(C) can be characterized as an optimization problem whose solutions are area minimizing boundaries contained in R N \ C, trapped in between two parallel hyperplanes, and intersecting C orthogonally.Moreover, exterior isoperimetric sets E v for C with v large can be fully "resolved as" (i.e., expressed as small diffeomorphic deformations of) the union of a large sphere of volume v missing a spherical cap and of an optimizing boundary in R(C).Obtaining this resolution result requires the introduction a new ε-regularity criterion operating at mesoscales and interpolating between the local Allard's regularity theorem [1], based on the analysis of blowups, and the "at infinity" regularity theorems of Allard-Almgren [2] and Simon [15], based on the analysis of blowdowns.
Coming back to the case of convex bodies, we are left to consider the case when 1 ≤ d * (C) ≤ N − 2 (and thus, with N ≥ 3).In this case we do not expect R C (v) to have a finite limit as v → ∞, but rather to depend on v through a power law.This expectation is not completely confirmed, but it is definitely supported, by our second main result.
Theorem 1.3.Let C be a convex body in R N , N ≥ 3, with d * (C) ∈ {1, ..., N − 2}.Then there exists a positive constant C 0 depending on N and C such that It seems plausible to conjecture that the lower bound in (1.7) should be capturing the correct order of magnitude of R C (v).In other words, it seems plausible that for every C as in Theorem 1.3 the limit should exist in (0, ∞), be amenable to be characterized as an optimization problem, and thus lead to extending the definition of isoperimetric residue to this class of obstacles.Resolving these issues would require fine geometric information on minimizers obtained by the application of an ε-regularity criterion analogous to [12].Given the considerable complexity of this problem even in the compact case, we leave this question for future investigations; see Remark 5.7 for more discussion.
We now discuss the organization of the paper and, in the process, we highlight some additional noteworthy statements proved in the paper but not included in Theorem 1.1 and Theorem 1.3.After setting our notation and recalling some basic facts concerning isoperimetric problems in Section 2, in Section 3 we collect several facts concerning the notion of asymptotic dimension: in particular, in Proposition 3.3 we describe the structure of sets with d * (C) ≤ N − 1.In Section 4 we introduce a constrained exterior isoperimetric profile I C,R (obtained by restricting competitors in I C to be contained in B R (0)), study its properties (Proposition 4.1), and then use those for proving Theorem 1.1.Finally, in Section 5 we discuss the problem of defining isoperimetric residues for unbounded convex sets with asymptotic dimension less than N − 2, and prove in particular Theorem 1.3, together with some properties of large volume exterior isoperimetric sets (see Theorem 5.2 and Proposition 5.5) that lay the groundwork for further analysis.
Acknowledgments: FM has been supported by NSF Grant DMS-2247544.FM and MN have been supported by NSF Grant DMS-2000034 and NSF FRG Grant DMS-1854344.MN has been supported by NSF RTG Grant DMS-1840314.NF has been supported by PRIN Project 2017TEXA3H.The research of MM was partially supported by GNAMPA and by the University of Parma via the project "Regularity, Nonlinear Potential Theory and related topics".

Notation and preliminary results about perimeter minimizers
In the following we shall denote by B R (x) the ball with center at x and radius R. If the center is at the origin we shall simply write B R .Given v > 0 we let B (v) (x) denote the ball of volume v centered at x ∈ R N and B (v) = B (v) (0).
Given E ⊂ R N of locally finite perimeter and a Borel set G we denote by P (E; G) the perimeter of E in G.The reduced boundary of E will be denoted by ∂ * E, while ∂ e E will stand for the essential boundary defined as where E (0) and E (1) are the sets of points where the density of E is 0 and 1, respectively.In the following, when dealing with a set of locally finite perimeter E, we shall always tacitly assume that E coincides with a precise representative that satisfies the property ∂E = ∂ * E, see [11,Remark 16.11].A possible choice is given by E (1) for which one may easily check that (2.1) We premise some lemmas.The first lemma is proved for instance in [7,Lemma 3.6] for N = 3, but the same statement (with the same argument) holds in any dimension.Lemma 2.1.Let C ⊂ R N be a convex body and let E ⊂ B R \C satisfy the following minimality property: there exists Λ ≥ 0 such that (2.2) Then E is equivalent to an open set, still denoted by E, such that ∂E = ∂ e E and hence Moreover, there exist c 0 = c 0 (N ) > 0 and r 0 = r 0 (N, Λ) ∈ (0, 1) independent of R and C, such that if x ∈ ∂E ′ , E ′ being a connected component of E, then 2) holds for all F ⊂ R N \ C, then the above density estimates holds also for We recall that a sequence {F n } of closed sets converges in the Kuratoswki sense (or locally in Hausdorff sense) to a closed set F if the following conditions are satisfied: (i) if x n ∈ F n for every n, then any limit point of {x n } belongs to F ; (ii) any x ∈ F is the limit of a sequence {x n } with x n ∈ F n .One can easily see that In particular, by the Arzelà-Ascoli Theorem any sequence of closed sets admits a subsequence which converge in the sense of Kuratowski.
For the simple proof of the next lemma see for instance [7,Remark 2.1].
Lemma 2.2.Let {C n } n be a sequence of closed convex sets.Then C n → C in the Kuratowski sense if and only if χ Cn → χ C pointwise almost everywhere.In addition, C is convex.
The next lemma is also well known, for the proof see for instance [9, Lemma 5.1].
Lemma 2.3.Let C be a closed convex set with nonempty interior and F ⊂ R N \ C a bounded set of finite perimeter.Then The following result, see [13,Lemma 2.1], is a simplified version of a nucleation lemma due to Almgren [3,VI.13];see also [11,Lemma 29.10].
Lemma 2.4.There exists a constant c(N ) > 0 such that if E ⊂ R N is a set of finite perimeter and finite measure, then, setting Q := (0, 1) N , we have Finally we conclude with a useful construction that allows one to locally dilate sets of finite perimeter with a controlled change in volume and perimeter.
Lemma 2.5.There exist ε(N ), c(N ) > 0 with the following property.Let E ⊂ R N be a set of finite perimeter with |E| < m and B R a ball such that The proof of Lemma 2.5 can be deduced from the proof of [6, Theorem 1.1].
We conclude this section with the relative isoperimetric inequality outside convex sets due to Choe, Ghomi and Ritoré together with the characterization of the equality case.
Theorem 2.6.Let C ⊂ R N be a convex body.For any set of finite perimeter E ⊂ R N \ C we have Moreover, if equality holds in (2.4), then E is a half ball supported on a facet of C.
The proof of (2.4) and the characterization of the equality case when C is a C 2 convex set has been established in [4], while the characterization of the equality case for general convex sets has been obtained in [9].

Asymptotic dimension of a convex body
In this section we discuss various properties of the asymptotic dimension d * (C) of a convex body C introduced in (1.6).We also recall the definition (1.5) of the recession cone C ∞ of C. Remark 3.1 (Convex bodies with zero/full asymptotic dimension).Note that if C is a bounded convex body then d * (C) = 0. Conversely, if d * (C) = 0 and we fix x ∈ C, then λ(C − x) must converge to {0} as λ → 0. Therefore we have that the recession cone C ∞ = {0} and thus C is bounded, see [14,Theorem 8.4].Note also that if d * (C) = N it is clear from the definition that C contains open balls of arbitrarily large radius.The converse is also true.Indeed, if B rn (x n ) ⊂ C and r n → ∞, by testing the definition with r Remark 3.2.Note that if C is an unbounded convex body, testing (1.6) with the constant sequence x n = x ∈ C and any λ n → 0, we deduce that with the inequality being possibly strict.Note also that d * (C) can also take any value between 1 and In the next proposition we deal with the structure of a convex body C such that 1 ≤ d * (C) ≤ N − 1.To this aim, if Z ⊂ R N is a subspace, we denote by p Z the orthogonal projection on Z. Furthermore, Z ⊥ will denote the subspace orthogonal to Z and if z ∈ R N , we will denote by z ⊥ the hyperplane orthogonal to z through the origin.
where ri(•) stands for the relative interior of its argument.Next, since ri(D) ⊂ p z ⊥ (C) and z ∈ C ∞ , we may choose T n ≥ 0 large enough and w 0,n ∈ z ⊥ such that for all t ≥ T n where in the last inclusion we used the fact that C t 1,n ⊂ C t 2,n .By the convergence w i,n → y i , we can restrict to a further subsequence, which we do not notate, that satisfies ⊂ z ⊥ are linearly independent, the elements of V are the vertices of a (d * (D) + 1)-dimensional simplex S, which belongs to C ′ since C ′ is convex.Since t 1,n z + x n ∈ C, we have thus shown (3.1).
For the reverse inequality we recall that by the definition of D, Up to a further subsequence, the right hand side of the previous equation converges in the Kuratowski sense to span {z} + D ′ for some closed convex To finish the proof it remains to show (a) and (b), which we do by strong induction on 1 ≤ d * (C) ≤ N − 1.For the base case when d * (C) = 1, item (a) follows from to the containment (3.3).Moreover, item (e) implies that the associated D := cl (p z ⊥ (C)) ⊂ z ⊥ satisfies d * (D) = 0, and so by Remark 3.1, D is a bounded convex body contained in z ⊥ , which is (b).
Suppose now that (a) and (b) are true for any convex body

Rigidity in the Choe-Ghomi-Ritoré comparison theorem
In this section we prove various properties of the exterior isoperimetric profile I C of a convex body, and then prove Theorem 1.1.We begin with the following proposition, where we consider a constrained version of the exterior isoperimetric profile.Then I C,R is a locally Lipschitz function in (0, |B R \ C|) and for a.e.v > 0, , where E v is any minimizer of the problem (4.1).Moreover for all v > 0 (4.2) lim Furthermore, assuming, up to a translation, that 0 ∈ ∂C, there exist positive constants Λ 0 , d 0 , r 0 , c 0 , and an integer I 0 ∈ N, all depending on N (but not on C), such that for all has at most I 0 connected components each of them with diameter less than v 1 N d 0 and for every connected component Finally, for any such minimizer we have Proof.Note that if R ≥ R 0 for any convex set C such that 0 ∈ ∂C there exists a set of finite perimeter E ⊂ B R \ C with |E| = 1.Indeed, C is contained in a half space and R 0 is strictly bigger than the radius of a half ball of volume 1.We divide the proof in three steps.
Step 1: The Lipschitz continuity and the representation formula for the derivative of I C when C is a bounded convex body are well known facts.The proof of the same properties for I C,R is similar, see for instance Steps 3 and 4 of the proof of Theorem 1.2 in [9].The proof of (4.2) is immediate.Indeed we clearly have For the opposite inequality it is enough to observe that any competitor for I C (v) can be approximated in (relative) perimeter by bounded sets of the same volume.
Step 2: The argument needed to prove that there exists Λ 0 such that (4.3) holds is similar for instance to the one in Step 6 of the proof of [7,Theorem 3.2] with some modifications due to our particular setting.We reproduce it here for the reader's convenience.To this aim, by rescaling, it is enough to show that there exists Λ 0 such that for every convex body C, any v > 0 and any R ≥ R 0 , every minimizer for the penalized problem with volume 1.
Let us suppose then for contradiction that there exist a sequence Λ j → ∞, R j ≥ R 0 , C j , v j ∈ (0, +∞) and minimizers E j,Λ j for (4.5) (with Λ 0 , R, v and C replaced by Λ j , R j , v j and C j , respectively) which do not have volume 1.We observe that necessarily |E j,Λ j | < 1, since otherwise we could contradict the minimality by cutting E j,Λ j with a hyperplane not intersecting v Thus by Lemma 2.4, there exists a constant c(N for some z j ∈ Z N and for every j.Therefore, up to a subsequence (not relabelled), we may assume that χ E j,Λ j −z j → χ E a.e., with E of finite perimeter and |E| ≥ c(N ).
We claim that there exist x ∈ ∂ * E and r > 0 such that C j , for all j sufficiently large.
To see this note that, up to a not relabelled subsequence, we may assume that C j → K in the sense of Kuratowski, for a suitable closed convex set K.Moreover, by Lemma 2.2 we have that χ K j → χ K almost everywhere.In particular, for a.e.x ∈ R N we have χ E (x)χ K (x) = lim j χ E j,Λ j −z j (x)χ K j (x) = 0, i.e., E ⊂ R N \ K. Observe also that, up to a further not relabelled subsequence, we may assume that −z j + B R j converge in the Kuratowski sense to K, where K can be R N or a half space containing ).Thus, from the Kuratowski convergence (4.8) follows.
Arguing as in Step 1 of Theorem 1.1 in [6], given 0 < ε < ε(N ), where ε(N ) is as in Lemma 2.5, we can find a ball B r (x 0 ) ⊂ B r (x) such that Therefore, for j sufficiently large, we have where by (4.8), B r ( . We now apply Lemma 2.5 to find a positive sequence {σ j } and a sequence { E j } such that E j \ B r (x 0 + z j ) = E j,Λ j \ B r (x 0 + z j ) and satisfying | E j | < 1 and From these inequalities, recalling (4.6) and that |E j,Λ j | < |E j | < 1, we then get N − Λ j c(N )r N < 0 for j large, as Λ j → ∞.This contradicts the minimality of E j,Λ j , thus proving (4.3).
Step 3: We finally show the last part of the statement.Assume now R ≥ R 0 and v > 0 and let E a minimizer for the problem defining I C,R v 1/N (v).From (4.3) it follows that Appealing to Lemma 2.1, the conclusions of which do not depend on R and v −1/N C, we obtain c 0 (N ) > 0 and r 0 depending only on N, Λ 0 and thus only on N , such that for every connected component Since |v −1/N E| = 1, this implies the existence of I 0 ∈ N such that the number of connected components of v −1/N E is at most I 0 .Moreover, the density estimate (4.9) also implies by a standard argument the existence of d 0 such that diam (E ′ ) ≤ d 0 for any component E ′ .Finally, the estimate (4.4) follows from the Λ-minimality property (4.3) by a standard first variation argument.
We now prove a semicontinuity property of the exterior isoperimetric profile with respect to the Kuratowski convergence of the set C. Lemma 4.2.Let {C n } be a sequence of convex bodies converging in the Kuratowski sense to a convex body C. Then for all v > 0 we have Proof.Without loss of generality we may assume that 0 ∈ int(C).Let I C,R be the constrained exterior isoperimetric profile defined in (4.1).For any ε > 0 sufficiently small let E ⊂ B R \ C be a minimizer of the problem defining I C,R ((1 + ε) N v).By the local Hausdorff convergence we have that for n large enough.Set E n := E \ (1 + ε)C n and observe that We now let We conclude the proof by observing that

Proof.
In what follows we denote by B ′ r (x) the intersection B r (x) ∩ {x N = 0}.We divide the proof in two steps.
Step 1: We start with the case d := dim C ∞ = N −1.Without loss of generality we may assume that C ∞ ⊂ {x N = 0} and that e 1 belongs to the relative interior of C ∞ .Let C ′ = C ∩ {x N = 0} and note that C ∞ ⊂ C ′ .Therefore there exists r > 0 such that B ′ 2r (e 1 ) ⊂ C ∞ .Consider now the sequence of balls B ′ nr (ne 1 ) and for any n denote by ϕ n : ϕ n .
Thus, thanks to (4.10) we have that Therefore it is easily checked that the sets C − (ne 1 , ϕ n (ne 1 )) converge in the Kuratowski sense, up to a subsequence, to a convex set K ⊂ {x N ≤ 0} such that ∂K ⊂ {x N = 0}.Let us now fix v > 0 and denote by r v the radius of a half ball of volume v.For every n let r n be the radius of the ball centered at x n = (ne 1 , ϕ n (ne 1 )) and such that Then, recalling that by (4.11) the boundary of C − x n is flattening out, it follows that r n → r v and that , while the opposite inequality follows from Theorem 2.6.
Step 2: We assume now that d = N .Without loss of generality we may assume, up to a possible rotation and dilation that C ∞ has a unique tangent plane {x N = 0} at e 1 and that C ∞ stays above {x N = 0}.Let us now fix κ > 0 so large that B ′ Observe that for any x ′ ∈ B ′ n κ (ne 1 ) the above infimum is finite.Indeed, if for some point x ′ we had ϕ n (x ′ ) = −∞, then the half line {te N : t ≤ 0} would be contained in C ∞ , which is not possible.A similar argument shows also that if Indeed, assuming without loss of generality that the sequence x ′ n n , ϕn(x ′ n ) n converges to some point y = (y ′ , y N ) ∈ R n then necessarily y ∈ ∂C ∞ and thus y N ≥ 0. Since ϕ n is a convex function, as before we have where where the last inequality follows from the fact {x N = 0} is tangent to C ∞ at e 1 .Therefore, recalling (4.12) and (4.13) we get that Fix R > 0. From the previous estimate we have that for all κ > 0 lim sup )) = 0. Then conclusion then follows exactly as in the final part of Step 1.
We are finally ready to prove Theorem 1.1.
Up to a subsequence, we may also assume that C − x n → K for some convex body K.In particular, since for any λ > 0 and n sufficiently large we have that K max ⊂ λK for every λ > 0, and thus In turn, by Lemma 4.3 we have that I K = I H . Hence, the lower semicontinuity property stated in Lemma 4.2 for all v > 0 we have that I C (v) ≤ I K (v) = I H (v), while the opposite inequality follows by the isoperimetric inequality (2.4).

Assume now d
Without loss of generality we may assume 0 ∈ ∂C.Given any diverging sequence v n → +∞, it will be enough to show that (4.14) lim inf Without loss of generality we may assume that the above lim inf is a limit.By Proposition 4.1, we may find Λ 0 > 0 and R n → +∞ such that Let E n be a minimizer of (4.16).Again by Proposition 4.1, passing possibly to a not relabelled subsequence, there exist κ ∈ N and d 0 > 0 such that for n sufficiently large each E n has κ connected components E n,i and each of them has diameter less than d 0 .We claim that for all i = 1, . . ., κ (4.17) To this aim fix i and assume P (E n,i ; ∂(v Hence, for any ε > 0 and n large enough, setting (K i ) ε := {x : dist(x, K i ) ≤ ε}, we have where the last inequality follows from the containment (holding for n large by Kuratowski convergence) and the fact that both sets are convex.Since P (B d 0 ∩ (K i ) ε ) → 0 as ε → 0, (4.17) follows.
In turn, by the isoperimetric inequality, and (4.14) then follows, recalling (4.15) and (4.17).This concludes the proof of the theorem.Proof.By Theorem 1.1, we have I C (v) = I H (v). On the other hand, by the characterization of the equality case in Theorem 2.6, taking into account that C is strictly convex, we have Remark 4.5 (Minimizers and generalized minimizers).Note that if C is a convex cylinder, then there exists a minimizer for the problem defining I C (v) for all v > 0. Indeed, by Proposition 4.1 any minimizer E R of I C,Rv 1/N (v) for R ≥ R 0 has at most I 0 connected components of diameter at most d 0 v 1/N .Letting R → ∞ and availing ourselves of the translation invariance outside a cylinder and the convergence of I C,R v 1/N to I C (v), we see that up to a subsequence and translations, the E R 's converge to a minimizer E of I C (v) as R → ∞.The veracity of (4.3) among any F ⊂⊂ R N \ C follows from noticing that for any B R (0) containing E R ∪ F (4.3) is satisfied by E R and then passing to the limit as R → ∞.
Finally, we remark that for general convex sets C one could prove the existence of generalized minimizers for the problem I C (v) in the following sense: Let E R be as before, with R → +∞, and pick z R ∈ E R .Then, up to a subsequence, we may assume that the Kuratowski sense, with K ∞ being a (possibly lower dimensional) convex cylinder, see [10,Lemma 3.1].It could be possible to show that E ∞ is a minimizer for the "asymptotic problem" I K∞ (v).The set E ∞ can be regarded as a generalized minimizer for I C (v) capturing the behaviour of (suitable) minimizing sequences.Note that d * (K ∞ ) ≤ d * (C).
5. The order of isoperimetric residue for unbounded convex bodies with asymptotic codimension larger than 2 The main goal of this section is providing a proof of Theorem 1.3.In fact, we shall prove various other results that seem potentially useful for future investigations too.Specifically, as already explained in the introduction and as indicated by [12] for the case d * (C) = 0, the question of understanding the behavior of R C (v) as v → ∞ is closely related to the description of minimizers of I C (v) as v → ∞.Since such minimizers may fail to exist, here we explore the idea of using minimizers of the constrained isoperimetric problems I C,R v 1/N already used in the previous section.In particular, in the following lemmas, we obtain basic information on the shape of such minimizers; see, in particular, Theorem 5.2 and Proposition 5.5.
Before moving forward with the above program we introduce some additional notation and terminology.In the following we denote by c(N, C) a positive constant depending only on C and N whose value may change from line to line or even within the same line.Moreover, given a set of finite perimeter E, with v = |E|, we denote the isoperimetric deficit and the Fraenkel asymmetry of E by (5.1) δ iso (E) := P (E) respectively.Finally, in this section, if A and B are positive quantities associated with a fixed convex body C, by A B we mean that there exists a constant c(N, C) > 0 such that A ≤ c(N, C)B.
The conclusion then follows by iterating the argument.
The following theorem contains Theorem 1.3 as a particular case.
Proof.Throughout the proof we will use the fact that, by Proposition 3.3, there exists a subspace Z of dimension d * (C), such that cl (p Z ⊥ (C)) is bounded and For each v > 0, let us now choose R(v) ≥ R 0 such that provided that v ≥ 1, where in the last inequality we used (5.6).From this estimate, (5.7) and the fact that there are at most I 0 connected components, we have Now we show that up to increasing the value of v 0 obtained in step two, any minimizer where the E i 's are the connected components of v −1/N E, I 0 i=1 |E i | = 1 and the last inequality follows arguing as in (5.9) 2 .But for all i, |E i | ≥ c 0 r N 0 , where r 0 is as in Proposition 4.1, and so by the concavity of t → t (N −1)/N , I 0 ≥ 2 is impossible for large enough v. Therefore, recalling (5.7) and (5.8), and arguing as in (5.9) we have Hence, the last two inequalities in (5.3) follow from the above inequality and (5.6).In turn, (5.4) follows from these inequalities and from the quantitative isoperimetric inequality proved in [8].
We now prove the first inequality in (5.3).By Lemma 5.1 it suffices to estimate R C from below when C is a convex cylinder of the form Z + D, with D ⊂ Z ⊥ bounded and Z a subspace of dimension d * (C).With no loss of generality we may assume that Z = {x ∈ R N : x = (x 1 , . . ., x d * (C) , 0 . . ., 0)} and that the cube In the following we will denote a point in R N as x = (x ′ , y ′ , x N ), where x ′ ∈ Z, y ′ = (x d * (C)+1 , . . ., x N −1 ).We will simply attach a large ball to Z + D, utilizing Q to bound from below R Z+D (v).For every r, consider the ball B r (−re N ).By the choice of Q, we estimate for r sufficiently large.By a similar argument we may estimate (5.12) Up to changing the constant c(N, C) as necessary, we combine (5.11) and (5.12) to obtain r for large enough r, where all the constants above may change from line to line.
, we have proven the lower bound in (5.3).
In the next lemma we complement the upper bound given in (5.2) by a corresponding lower bound.
In particular, Proof.By Theorem 5.2 (and assuming without loss of generality 0 ∈ ∂C), given v ≥ v 0 and R ≥ R 0 there exists a connected minimizer E v of the problem defining . By the containment of C in Z + D, we may write Now by the diameter bound on E v and boundedness of D, we know that We conclude with the following proposition, which provides an estimate on the proximity to balls of large volume isoperimetric sets.For simplicity we assume C to be a convex cylinder, as in this case we can ensure the existence of minimizers for the relative isoperimetric problem (see Remark 4.5).Given two compact sets K 1 , K 2 , we denote here by hd (K 1 , K 2 ) = max max .for a suitable r 0 = r 0 (N ) > 0. Suppose now that for some x ∈ cl(∂E \ C), (5.20) h := hd (∂E \ C, ∂B (v) (x 0 )) = dist (x, ∂B (v) (x 0 )) > 0 ; we will handle the other case for computing h in (5.24).Then B h (x) ∩ ∂B (v) (x 0 ) = ∅, so due to (5.19) and the quantitative isoperimetric inequality, we get
and examples where this inequality is strict are easily constructed.Having completely discussed the rigidity problem for the CGR-comparison theorem, we move to the problem of understanding which information on the convex body C is stored in the behavior of I C (v) as v → ∞, and what can be said, should they exist, about large volume exterior isoperimetric sets of C, i.e., about minimizers E v of I C (v) as v → ∞.Theorem 1.1 plays an interesting role in setting up these problems.Indeed, Theorem 1.1 states that for a convex body C with d * (C) ∈ {N − 1, N } there is no geometric information (in addition to d * (C) ∈ {N − 1, N }) that can be found by studying I C for v large, as I C must then be identically equal to I H .At the same time, when d * (C) ≤ N − 2, Theorem 1.1 invites us to investigate what information about C may be stored in the behavior of the quantity in the Kuratowski sense to cl (p z ⊥ (C)) as t → ∞, and (e) d * (cl (p z ⊥ (C))) = d * (C) − 1. Remark 3.4.Let C be an unbounded convex body and assume that Z is a subspace such that (a) and (b) of Proposition 3.3 hold.Then dim Z = d * (Z + cl (p Z ⊥ (C))) ≥ d * (C), where we used the fact that d * (•) is monotone with respect to the inclusion.Therefore one could equivalently define d * (C) as the smallest k ∈ N such that there exists a subspace Z with dim Z = k such that p Z ⊥ (C)) is bounded and C ⊂ Z + cl (p Z ⊥ (C)); if no such subspace exists, then d * (C) = N .Proof of Proposition 3.3.We first prove (c)-(e) and then (a) and (b).By Remark 3.1, C is unbounded and therefore C ∞ is nonempty, and so we can fix some z ∈ C ∞ .By the definition of C ∞ , x + αz ∈ C for all x ∈ C and α ≥ 0. From this we deduce the nested property (c) of {C t }.Item (d) then follows from the fact that any sequence of increasing, closed, convex sets converges in the Kuratowski sense to the closure of the union of its elements and observing that the closure of such a union coincides with cl (p Z ⊥ (C)).Moving on to (e), let us set D = cl (p z ⊥ (C)) for brevity.We begin by proving that d * (D) ≤ d * (C) − 1 .(3.1) Let x n ∈ D and λ n → 0 be such that λ n (D − x n ) converges in the Kuratowski sense to D ′ , where dim D ′ = d * (D).Then, taking into account 0 ∈ D ′ , there exist linearly independent vectors y 1 , . . ., y d * (D) ∈ D ′ .By the convergence of λ n (D − x n ) to D ′ , we can therefore choose a sequence ε n → 0 and w i,n ∈ z ⊥ such that a convex body with d * (C) =: k + 1 > 1 and z ∈ C ∞ , we apply (c), (d) and (e) to C and to the (N − 1)-dimensional convex body D := cl (p z ⊥ (C)) ⊂ z ⊥ with d * (D) = k ≥ 1 to obtain (3.4) C ⊂ span {z} + D .By the induction hypothesis applied to D, we may obtain N − 1 orthonormal vectors {z 2 , . . ., z 1+d * (D) , y 1 , . . ., y N −d * (D)−1 } ⊂ z ⊥ such that setting Z ′ = span {z 2 , . . ., z 1+d * (D) } ⊂ z ⊥ and Y = span {y 1 , . . ., y N −d * (D)−1 } ⊂ z ⊥ , we have (3.5)D ⊂ Z ′ + cl (p Y (D)) and (3.6) cl (p Y (D)) is bounded .Therfore, by (3.4)-(3.5),setting Z := span {z, z 2 , . . .z 1+d * (D) }, we have C ⊂ Z + cl (p Y (D)) = Z + cl (p Y (cl (p z ⊥ (C)))) .Since d * (C) = d * (D) + 1 and cl (p Y (cl (p z ⊥ (C)))) = cl (p Y (C)), we have shown (a).The boundedness of cl (p Y (C)), which is (b), follows from the induction hypothesis (3.6).

Lemma 4 . 3 .
and using Proposition 4.1 to let ε → 0 and then R → ∞.Let C ⊂ R N be a convex body and let H be any half space.If dim C ∞ ≥ N − 1, then I C = I H .

1 κ(e 1
) is contained in the projection of C ∞ onto {x N = 0}.Consider the balls B ′ n κ (ne 1 ) and the functions

Corollary 4 . 4 .
Let C be a strictly convex body with d * (C) ≥ N − 1.Then, for every v > 0 the relative isoperimetric problem (1.1) does not admit a solution.

Theorem 5 . 2 (
Lower and upper bounds for R C ).Let N ≥ 3 and let C ⊂ R N be a convex body with 1 ≤ d * (C) ≤ N − 2. Assume without loss of generality that 0 ∈ ∂C.Then, there exists v 0 ≥ 1 depending on C and N , such that for all R

Lemma 5 . 3 (
Estimate from below by convex cylinders).Let N ≥ 3, C ⊂ R N be a closed convex body with 1 ≤ d * (C) ≤ N −2, and Z be a d * (C)-dimensional subspace as in Proposition 3.3 with D := cl (p Z ⊥ (C)).Then there exist v 0 > 0 and c(N, C) such that (5.13)

Proposition 5 . 5 (Remark 5 . 6 . 1 P
Uniform convergence to balls).Let 1 ≤ k ≤ N − 2 and let C = Z + D, with Z a k-dimensional subspace and D ⊂ Z ⊥ a compact (N − k)-dimensional convex set.Then there exists v 1 = v 1 (N, C) > 0 such that if E is a minimizer for the problem defining I C (v), with v ≥ v 1 , and if the ball B (v) (x 0 ) is optimal for the definition (5.1) of A(E), then hd (∂E \ C, ∂B (v) (x 0 )) v 1/N (diam C ∩ ∂E) k/(2N ) v (N −1)/(2N 2 ) v −(N −1−k)/(2N 2 ) .(5.18)It should be noted that for general convex sets C, with 1 ≤ d * (C) ≤ N − 2, the conclusion of Proposition 5.5 applies to the generalized minimizers introduced in Remark 4.5, with k = d * (C).Proof of Proposition 5.5.Let v ≥ v 0 , where v 0 is from Theorem 5.2, so that any minimizer E is connected.Recall also that by Proposition 4.(E; R N \ C) ≤ P (F ; R N \ C) + Λ 0 v − 1 N |F | − |E| ∀F ⊂ R N \ C .In turn, this implies by Lemma 2.1 together with a rescaling argument (see also the proof of Proposition 4.1) that(5.19)min {|E ∩ B r (x)|, |B r (x) \ E|} ≥ c 0 r N for all 0 < r ≤ r 0 v 1 N .21) On the other hand, since d * (C) = k, (5.4) reads(5.22)δ iso (E)(diam C ∩ ∂E) k v (N −1)/N v −(N −1−k)/N ,in which case the minimum in (5.21) must achieved by h if v ≥ v 1 and we choose v 1 ≥ v 0 large enough.Combining (5.21) and (5.22), we arrive at(5.23)h ≤ C(N )δ 1/(2N ) iso (E)v 1/N v 1/N (diam C ∩ ∂E) k/(2N ) v (N −1)/(2N 2 ) .Since diam E ≤ d 0 v 1/N by (5.6), dividing (5.23) by v 1/N yields (5.18) in the case that (5.20) holds.Conversely, if there exists x ∈ ∂B (v) (x 0 ) such thath := hd (∂E \ C, ∂B (v) (x 0 )) = dist (x, ∂E \ C) > 0 , (5.24) then either B h (x) ⊂ E or B h (x) ⊂ R N \ E.An analogous argument as in the previous case leads again to (5.23) and thus to (5.18).Remark 5.7.Improved estimates on the Hausdorff distance of ∂E to the large ball could be used to determine an upper bound for R C (v).In a nutshell, one would need to prove Lemma 5.5 with ∂E in (5.18) instead of ∂E \ C. Let us explain why these estimates should hold and then how the argument would go.Assume for simplicityC = Z + D with dim Z = k = d * (C) ≤ N − 2, dim D = N − d * (C) and D ⊂ Z ⊥ .The estimate (5.18) with ∂E would hold if we knew for example that minimizers E never "envelop" C; that is, there are no slices C z = C ∩ (z + D), with z ∈ Z, such that ∂C z ⊂ ∂E.If this were true (and it seems like it should be -why should a minimizer envelop the obstacle?),then every point in ∂E would be close to a point in ∂E \ C and thus (5.18) would hold with ∂E.Thus we would have hd (∂E, ∂B (v) (x 0 )) v 1/N (diam C ∩ ∂E) d * (C)/(2N ) v (N −1)/(2N 2 ) .