Nonexistence of isoperimetric sets in spaces of positive curvature

For every $d\ge 3$, we construct a noncompact smooth $d$-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below $1$. We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in $\mathbb R^d$. The examples we construct have nondegenerate asymptotic cone. The dimensional constraint $d\ge 3$ is sharp. Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed in nonnegatively curved spaces with nondegenerate asymptotic cones isoperimetric sets with large volumes always exist. This is the first instance of noncollapsed nonnegatively curved space without isoperimetric sets.


Introduction
The aim of this paper is to prove the following two nonexistence results for isoperimetric sets (see Definition 2.1).
Theorem 1.1.For every d ≥ 3, there exists a smooth complete d-dimensional Riemannian manifold (M, g) with strictly positive sectional curvature everywhere and with nondegenerate asymptotic cone 1 for which the following hold.
(1) If v < 1, there are no isoperimetric sets of volume v in M ; (2) If v > 1, there exists an isoperimetric set of volume v in M .
Theorem 1.2.For every d ≥ 3, there exists a closed strictly convex set C ⊆ R d with nonempty interior, smooth boundary, and with nondegenerate asymptotic cone 2 for which the following hold.
(1) If v < 1, there are no (relative) isoperimetric sets of volume v in C; (2) If v > 1, there exists a (relative) isoperimetric set of volume v in C.
Let us remark that with our technique it should be possible to produce also examples of manifolds and convex sets satisfying Theorems 1.1 and 1.2 with asymptotic cone that is not nondegenerate.
1.1.Overview of existence results for isoperimetric sets.Our result complements the vast literature about the existence, under suitable assumptions, of isoperimetric sets in noncompact spaces (the existence in compact spaces is standard nowadays, see, e.g., the proof in [Mag12,Proposition 12.30]).The usual assumptions are some form of nonnegativity of the (sectional, Ricci, or scalar) curvature and/or some rigidity of the structure of the space at infinity.Under a (possibly negative) bound from below on the Ricci curvature, and a uniform lower bound on the volume of unit balls, one can show that, for a given volume, isoperimetric sets exist either in the space itself or the mass may split in a finite number of isoperimetric sets in its limits at infinity [RR04;Nar14].See also the generalizations to nonsmooth spaces in [ANP22].
The weak existence results mentioned in the previous paragraph are used in [MN16] to deduce, e.g., the existence of isoperimetric sets for all volumes if the Ricci curvature is nonnegative and all the limits at infinity are Euclidean.
After the works in [LRV22; Ant+22; Ant+23a], it is known that if the sectional curvature is nonnegative 4 and the asymptotic cone (cf., Eq. (2.1)) is nondegenerate, isoperimetric sets exist for all sufficiently large volumes.Analogous results, with slightly stronger assumptions on the asymptotic cones, are obtained also assuming only the nonnegativity of the Ricci curvature (in place of the nonnegativity of the sectional curvature).In the particular case of cones with nonnegative Ricci curvature, isoperimetric sets exist for all volumes and they are balls centered at one of the tips of the cone [LP90;MR02].
The problem for 2-dimensional spaces is completely settled in [Rit01; AP23]: in nonnegatively curved surfaces existence holds for every volume.
The existence of isoperimetric sets for 3-dimensional manifolds with nonnegative scalar curvature is known for asymptotically flat manifolds [CCE16, Proposition K.1], [Shi16].The result holds also if one assumes that the manifold is asymptotic (in a quantitative sense) to a cone with nonnegative Ricci curvature [CEV17], without any additional assumption on the curvature of the manifold.See also [EM13b;EM13a] for related results, and the recent survey [BF23].
In the works [LRV22; Ant+23b], a number of techniques are developed to deal with the possible lack of isoperimetric sets for certain volumes to show the concavity of the isoperimetric profile for nonnegatively curved spaces (in the compact case this difficulty does not arise [BP86, (iii) at page 483],[SZ99; Kuw03; Bay04]).The examples constructed in this paper tell us that these difficulties are unavoidable.1.2.Strategy of the construction and structure of the paper.In Section 2 we recall the basic notions and some results about the isoperimetric problem in nonnegatively 4 Here it is sufficient to assume that the space is an Alexandrov space with nonnegative sectional curvature.Among the spaces satisfying this assumption we find: Riemannian manifolds with nonnegative sectional curvature, convex bodies in Euclidean spaces, and their boundaries.
curved Alexandrov spaces.The novelty of this section is Lemma 2.6: whenever the space has a nondegenerate asymptotic cone, if all limits at infinity are cones, then there is a threshold for the existence of isoperimetric sets (i.e., for v < v 0 there is no isoperimetric set with volume v and for v > v 0 there is).To this end, we devise a robust proof of the concavity of the isoperimetric profile (to the power d d−1 ) that might be of independent interest (see Lemma 2.5 and Remark 2.2).
In Section 3, that is the core of the paper, we construct the convex set C which, after an approximation procedure performed in Section 4, will satisfy the properties of Theorem 1.2.The manifold satisfying the properties mentioned in Theorem 1.1 is obtained considering the boundary ∂C.
Let us quickly describe here the main features of our construction, we refer the reader to Section 3.1 for a more in-depth presentation and for a picture.
Consider a closed convex cone Σ ⊆ R d contained in the half-space {x ∈ R d : x • e 1 ≥ 0}.For a convex decreasing function φ : R → (0, ∞) with φ(+∞) = 0, we define C as A rather delicate choice of φ (so that φ ′ is pointwise small compared to φ) produces a convex set C without isoperimetric sets with small volume.The idea is that one limit at infinity of C is the cone Σ × R, which has density strictly lower than any tangent cone at a point of C.This already implies that if there were isoperimetric sets for small volumes v in C, then these sets would escape to infinity as v → 0, see [LRV22, Theorem 6.9].The crux of the argument is to upgrade this observation to the nonexistence of isoperimetric sets for small volumes.
The proof is easier to follow for C (i.e., for Theorem 1.2) than for ∂C (i.e., for Theorem 1.1).To avoid repeating many arguments, we give the full proofs for C and we emphasize only the differences in the proofs for ∂C.
Finally, in Section 5 we show how the results in the previous sections imply our two main theorems.Notable examples of nonnegatively curved Alexandrov spaces are: complete Riemannian manifolds with nonnegative sectional curvature; closed convex sets with nonempty interior in the Euclidean space, endowed with the Euclidean distance; their boundaries endowed with their geodesic distance; metric cones on smooth compact Riemannian manifolds with sectional curvature ≥ 1 (see [BBI01, Section 10.2]).In this paper we will be only concerned with closed convex sets with nonempty interior in the Euclidean space, and their boundaries, see [BBI01, Theorem 10.2.6] and [BBI01, Generalizations at p.359].The theory of Alexandrov spaces provides a unified approach to address both settings.The reader not familiar with the theory of Alexandrov spaces can safely assume that whenever we state a result for a nonnegatively curved Alexandrov space, we will apply it only for convex sets or their boundaries.

Acknowledgements
Let us recall that if the boundary of a closed convex set with nonempty interior in R d is smooth, then, when endowed with the metric tensor given by the pull-back of the Euclidean metric tensor, it is a smooth Riemannian manifold with nonnegative sectional curvature everywhere.
2.1.2.Asymptotic cones and limits at infinity.We are going to use the notion of pointed Gromov-Hausdorff (pGH) convergence (we refer the reader to [BBI01, Section 7 and Section 8.1]).For the aims of this paper, since we are concerned only with Alexandrov spaces embedded in the Euclidean space, the notion of Hausdorff convergence on compact sets would be equivalent.The following statement formalizes this intuition.
Moreover, the boundary ∂C n , endowed with its geodesic distance, converges in the pointed Gromov-Hausdorff sense to the boundary ∂C endowed with its geodesic distance.
Proof.Upgrading the Hausdorff convergence to the pointed Gromov-Hausdorff convergence is simple and we leave it to the reader.The convergence at the level of the boundaries is proven in [Pet97, Theorem 1.2].□ Let us define the notions of asymptotic cone and limits at infinity of a nonnegatively curved Alexandrov space (X, d).
The asymptotic cone (X ∞ , d ∞ ) of X is the limit (in the pointed Gromov-Hausdorff sense) for some r n → 0, and for some arbitrary point o ∈ X.The asymptotic cone does not depend on the choice of the rescaling factors r n or the point o ∈ X, it is itself a nonnegatively curved Alexandrov space, and it is a metric cone (see [BBI01, Definition 3.6.12]for the definition of metric cone).See, e.g., [BGS85, Exercises (a)-(e)].
For a d-dimensional nonnegatively curved Alexandrov space, we say that it has nondegenerate asymptotic cone if its asymptotic cone is d-dimensional.
We say that a metric space (Y, d Y ) is a limit at infinity of X if there exists a sequence of diverging points p n ∈ X, and 2.1.3.Perimeter and isoperimetric profile.As of today, a robust theory of functions of bounded variation and sets of finite perimeter has been developed in complete separable metric spaces endowed with a Radon measure [Mir03;AD14].Such a theory can be applied in the setting of d-dimensional nonnegatively curved Alexandrov spaces endowed with their Hausdorff measure H d , producing a well-behaved notion of perimeter of a set.We will not need the definitions here, but we refer the interested reader to [Mir03; AD14] for the general theory.
Given a nonnegatively curved Alexandrov space X and a Borel set E ⊂ X, we denote by Per X (E) the perimeter of E in X, defined as in [Mir03,Definition 4.1].This notion of perimeter agrees with the usual one in the following cases: (1) Assume that X is a closed convex set with nonempty interior in R d , for d ≥ 2.
For every Borel set E ⊂ X with Per X (E) < ∞, we have that E is a set of finite perimeter and, denoting with . Namely Per X (E) is the relative perimeter in the interior int(X).
(2) Assume that X is a smooth d-dimensional Riemannian manifold, for d ≥ 2. For every Borel set E ⊂ X with Per X (E) < ∞, we have that E is a set of finite perimeter and, denoting with Definition 2.1.For a nonnegatively curved Alexandrov space X, we denote by I X its isoperimetric profile, i.e., the function We say that a Borel set E ⊆ X is an isoperimetric set if Per X (E) = I X (|E|).We will always implicitly assume that isoperimetric sets are open [APP22, Theorem 1.4].
2.2.Known results on the isoperimetric profile and isoperimetric sets.Let us recall some standard facts in the theory of the isoperimetric problem for nonnegatively curved spaces.First of all, by scaling-invariance, if X is a d-dimensional metric cone then the isoperimetric profile satisfies (see [LP90, Theorem 1.1], [RR04, Proposition 3.1], or [MR02, Theorem 3.6] for the sharp isoperimetric inequality in convex cones in the Euclidean space, in Euclidean cones, or in nonnegatively Ricci curved cones).From now on, we will use the latter information without mentioning it.
The following lemma compares the isoperimetric profile of a nonnegatively curved Alexandrov space with the one of its asymptotic cone and limits at infinity.
Moreover, for all v > 0, we have The inequality I X ≤ I X ′ , for any limit at infinity X ′ , follows from the more general [ANP22, Equation (2.17The following lemma is a direct consequence of [Ant+23b, Lemma 4.20], see also [LRV22, Theorem 5.8] in the case of convex bodies.
Lemma 2.3.Given d ≥ 2, let X be a d-dimensional nonnegatively curved Alexandrov space with nondegenerate asymptotic cone.For all v > 0 the following holds: if there is no isoperimetric set with volume v in X, then there exists an isoperimetric set of volume v in a limit at infinity X ′ of X, and moreover Let us recall the following standard result on the diameter of isoperimetric sets.See [LRV22, Lemma 5.5] for the version on convex sets, and [Ant+23a, Proposition 4.23] for the general version on nonnegatively curved Alexandrov spaces.
Lemma 2.4.Given d ≥ 1, let X be a d-dimensional nonnegatively curved Alexandrov space with nondegenerate asymptotic cone, and assume that , where diam(E) denotes the diameter of E.

2.3.
The existence of a threshold.The goal of this section is to show that, under a mild assumption on the limits at infinity, there is a volumetric threshold for the existence of isoperimetric sets, see Lemma 2.6.
Our main tool is the following robust proof of the concavity of I X .Lemma 2.5.Given d ≥ 2, let X be a nonnegatively curved d-dimensional Alexandrov space.Let E ⊆ X be an isoperimetric set.For r ∈ R, define E r as5 There exists H ≥ 0, which agrees with the (constant) mean curvature of ∂E whenever X is sufficiently smooth and which agrees with achieves its maximum at r = 0 on (−∞, ∞).Moreover, for r ≥ 0, we have Proof.First of all, the existence of an isoperimetric set E implies that inf x∈X |B 1 (x)| > 0 (see [Ant+22, Proposition 2.18]); this assumption is necessary in some of the results we will cite.We will use repeatedly, without saying it explicitly, that the map r → Per X (E r ) is lower semicontinuous everywhere.In addition, by the co-area formula [Mir03, Proposition 4.2], we also have that d dr |E r | = Per X (E r ) for almost every r ∈ R and in the distributional sense, and Per X (E r ) ∈ L 1 loc (R).Let H ≥ 0 be a mean curvature barrier for E, as defined in [Ant+23b, Definition 3.6].The existence of such an H is guaranteed by [Ant+23b, Theorem 3.3], and the fact that H ≥ 0 is guaranteed by [Ant+23a, Proposition 3.1], since X has nonnegative curvature.The fact that H agrees with the usual mean curvature in the smooth case is observed in [Ant+23b, Remark 3.8].Finally, the fact that From [Ant+23b, Equation (3.37)] we get and Per X (E r ) = 0 for every r ≤ − d−1 H .When H = 0 we understand that − d−1 H = −∞.Thus, since r → Per X (E r ) is lower semicontinuous, we get that r → Per X (E r ) is continuous at r = 0. We will use this information repeatedly without saying it.Notice that, as a byproduct of Eq. (2.4), we also get that Per X (E r ) is locally bounded.
Following the proof of [Ant+23b, Equation (3.34)], but integrating on the slab From Eq. (2.5) we can infer Let us now show that the function in the statement is distributionally (weakly) decreasing for r ∈ [0, +∞).For r ∈ (0, ∞), it holds in the distributional sense that d dr Per X (E r ) where in the first inequality we used Eq.(2.6), and in the second inequality we used Eq.(2.4).We deduce that the function in the statement has a maximum at r = 0 when restricted to r ∈ [0, ∞).

|E|
and, for r > 0, the inequality Eq. (2.3) follows.□ Remark 2.1.The statement and proof of Lemma 2.5 work in the more general setting of noncompact metric measure spaces with synthetic nonnegative Ricci curvature.For the ease of readability and for consistency with the material of this paper we stated it for nonnegatively curved Alexandrov spaces.
Remark 2.2.Let X be a d-dimensional nonnegatively curved Alexandrov space.Let us see how Lemma 2.5 implies the concavity of and why it is a more robust method compared to the standard proof (see, e.g., [Ant+23b, Theorem 4.4], or [Bay04, Theorem 2.1] in the smooth setting).The proof starts by assuming, for each v 0 > 0, the existence of an isoperimetric set E with |E| = v 0 (possibly looking in a limit at infinity, and possibly considering simultaneously multiple limits at infinity, thus using [ANP22, Theorem 1.1]).
Adopting the same assumptions and notation of Lemma 2.5, consider the function that is, we have found a line that touches I X (v) The standard proof goes along the same scheme but replaces J(v) ≤ J(v 0 ) + (v − v 0 )γ with the weaker (and differential in nature) which is sufficient for the concavity but not for the rigidity argument employed in the proof of Lemma 2.6.Lemma 2.6.Given d ≥ 1, let X be a d-dimensional nonnegatively curved Alexandrov space with nondegenerate asymptotic cone.Assume that all its limits at infinity are cones.
Then there exists 0 ≤ v 0 < +∞ such that the following hold.
(1) For 0 < v < v 0 , there are no isoperimetric sets of volume v in X; (2) For v 0 < v, there exists an isoperimetric set with volume v in X.
Proof.If isoperimetric sets exist for every positive volume, v 0 = 0 works.From now on we assume that this is not the case.Let ϑ := inf{I X ′ (1) : X ′ is a limit at infinity of X} be the infimum of the perimeter of sets with unit volume in a limit at infinity of X. Denote with X ∞ the asymptotic cone of X.For all v > 0, as a consequence of Proposition 2.2, we have for any X ′ limit at infinity of X (we are using that all the limits at infinity are cones).In particular ϑ > 0.
Observe that is a decreasing continuous quantity that is ≤ ϑ everywhere; We show that the statement holds for this choice of v 0 .Notice that a posteriori v 0 < +∞ because we always have existence of isoperimetric sets for large volumes, see [Ant+23a, Item (1) of Theorem 1.2].
Assume that there is no isoperimetric set of volume ṽ > 0 in X.Then there exists an isoperimetric set with volume ṽ and perimeter I X (ṽ) in one of the limits at infinity of X, see Lemma 2.3.Therefore, taking Eq.(2.10) into account, we get I X (ṽ) = ϑṽ d−1 d .In particular ṽ ≤ v 0 .Take 0 < ṽ < v 0 and assume by contradiction that there is an isoperimetric set E ⊆ X of volume ṽ with perimeter equal to I X (ṽ) = ϑṽ in a neighborhood of ṽ, the constant H appearing in the statement of Lemma 2.5 coincides with H = I ′ X (ṽ) = d−1 d ϑṽ − 1 d and so d d−1 H Per X (E) For any r ∈ R, let E r be the set defined in Lemma 2.5; we have Hence, for any v > 0, since one can find r ∈ R so that This is a contradiction because we are assuming (see the beginning of the proof) that there is at least one volume v ∈ R so that there is not an isoperimetric set of volume v and we have shown above that in such case it must hold Remark 2.3.The authors believe that the assumption that the limits at infinity are cones is necessary in the statement of the previous lemma.In particular, it might be possible to construct a counterexample to the statement with the same method of this paper (choosing as Σ in Eq. (3.11) a convex set that is a compact perturbation of a cone).
For v = v 0 , under the same assumptions of Lemma 2.6, we expect an isoperimetric set to exist in certain cases and not to exist in others.
The boundary ∂Σ is smooth away from the origin.
Let C ⊆ R d+1 be the closed convex set with nonempty interior (here e 1 denotes the first element of the canonical basis of R d ) Representation of the boundary Notice that any slice of C at a fixed t = t 0 coincides with Σ λ for some λ > 0.
3.2.Properties of C. The asymptotic cone of C is Notice that the limit at infinity of C along the sequence of points (φ(t)e 1 , t) for t → ∞ is Σ × R. As a consequence it holds that I Σ×R ≥ I C (recall Proposition 2.2).Finally notice that all the limits at infinity of C are cones.
Observe that in the region {(x, t) : |x| > A 1 φ(0)}, the set C coincides with the cone the tip of the cone being 0 R d , − φ(0) φ ′ (0) .One can show this by looking at the sections of C and C ′ .For t 0 < 0, the sections of C and C ′ at t = t 0 coincide.To handle the sections for Therefore, the sections of C and C ′ at t ≥ 0 coincide with Σ when we consider only the region {x : |x| > A 1 φ(0)}.
Let us remark that Σ 1 × R is not isometric to Σ × R. One can show this observing that the density of Σ 1 at any of its points is strictly larger than the density of Σ at its tip; thus the density of Σ 1 × R at any point is strictly larger than the density of Σ × R at its tip.
3.3.Properties of the boundary ∂C.us now shift our attention to the boundary of C. Since it is the boundary of a convex set, it is in particular an Alexandrov space with nonnegative sectional curvature when endowed with its geodesic distance.The asymptotic cone of ∂C is the boundary of the asymptotic cone of C (see Lemma 2.1).Whenever we consider the boundary of a convex set, we implicitly assume that it is endowed with the geodesic distance.
Notice that the limit at infinity of ∂C along the sequence of points (φ(t)e 1 , t) for t → ∞ is ∂Σ × R (see again Lemma 2.1).As a consequence it holds that I ∂Σ×R ≥ I ∂C (recall Proposition 2.2).The section of ∂C at t = t 0 is ∂Σ φ(t0) .Analogously to what we observed above for C, also ∂C coincides with a cone (that is ∂C ′ ) in the region {(x, t) : |x| ≥ A 1 φ(0)}.
For d ≥ 3, we have that ∂Σ 1 × R is not isometric to ∂Σ × R (notice that for d = 2, ∂Σ 1 and ∂Σ are both isometric to R).This can be shown by observing that the tangent cone at any point of ∂Σ 1 splits at least one line6 , while the tangent cone at the tip of ∂Σ does not split any line.

3.4.
Diameter bound for isoperimetric sets in C and ∂C.The asymptotic cone of C contains Σ × [0, ∞).The asymptotic cone of ∂C contains ∂Σ × [0, ∞).These two remarks imply a lower bound on the volumes of the unit balls of the asymptotic cones of C and ∂C that is independent of the choice of φ.Therefore Lemma 2.4 tells us that there exist two constants and any isoperimetric set E ⊆ ∂C satisfies 3.5.Controlling the isoperimetric profile of C and ∂C.The following two lemmas (one for C and one for ∂C) provide a lower bound on the perimeter of a subset that is localized in space.
The function φ we have constructed satisfies all the properties in Eq. (3.12).
Assume by contradiction that E ⊆ C with |E| = 1 is an isoperimetric set, i.e., that Per C (E) = I C (|E|).
Space localization of the isoperimetric set E. As a first step, we show that we can assume that A 1 ≥ inf (x,t)∈E |x|.Since φ(0) = 1, in the region {(x, t) : |x| > A 1 } the set C coincides with the cone C ′ ⊆ R d+1 defined in Eq. (3.14).Let p ′ ∈ R d+1 be the tip of such cone.If inf (x,t)∈E |x| > A 1 , we replace E with a rescaled enlargement of itself as follows.For r > 0, denote with E r the set {z ∈ C : d E (z) ≤ r}.We can find r 0 > 0 and 0 < λ 0 < 1 such that the set E ′ defined as has volume 1 and inf (x,t)∈E ′ |x| = A 1 .Thanks to Eq. (2.3), we have so also E ′ is an isoperimetric set.By replacing E with E ′ , we gain the additional assumption A 1 ≥ inf (x,t)∈E |x|.Applying Eq. (3.15), we know that diam(E) ≤ A 2 .In particular, for some t 0 ∈ R, it holds that E ⊆ {(x, t) : t ∈ [t 0 , t 0 +A 2 ]}.Since we have reduced to the case inf (x,t)∈E |x| ≤ A 1 , the bound on the diameter implies that |x| ≤ A 2 + A 1 for all (x, t) ∈ E. Hence, we have Contradiction via Lemma 3.1.Both if t 0 < 0 or t 0 ≥ 0, we have that |φ ′ (t 0 )| ≤ h(φ(t 0 )) ≤ A 3 (d, A 1 + A 2 , α) for some 0 < α ≤ φ(t 0 ).Therefore we can apply Lemma 3.1 (with R = A 1 + A 2 ) and deduce Per C (E) > I Σ×R (1).This is a contradiction because Σ × R is a limit at infinity of C and therefore I C (1) ≤ I Σ×R (1).□ And we state the equivalent result on the boundary ∂C.

Smoothing the boundary of the construction
We prove two simple lemmas (one for a convex set and one for its boundary) which show that the nonexistence of isoperimetric sets is stable under an appropriate approximation procedure.The proofs use crucially the results of [AS23].
Proposition 4.1.Given d ≥ 2, let C ⊆ R d be a closed convex set such that ∂C does not contain any line (or, equivalently, C does not split a line) and its asymptotic cone C ∞ is nondegenerate.Assume that, for some v > 0, any Borel set E ⊆ C with |E| = v satisfies Per C (E) > I C (v) (i.e., there is no isoperimetric set with volume v).
There exists a strictly convex set C with smooth boundary so that Proof.For any compact set K ⊆ R d , a simple compactness argument tells us that there exists Therefore, with another compactness argument (which uses the lower semicontinuity of the perimeter along a sequence of Alexandrov spaces converging in the Gromov-Hausdorff sense [ABS19, Proposition 3.6]), we can find Let C ∞ be the asymptotic cone of C .Thanks to Lemma 2.4, we know that there exists a constant L = L(C ∞ , v) > 0 such that if C is a convex set with asymptotic cone C ∞ , then any isoperimetric set in C with volume ≤ v has diameter below L.
Let (K n ) n∈N be a sequence of compact sets such that ∪ n∈N K n = R d is a locally finite covering, and for any set E ⊆ R d with diameter below L there is n 0 = n 0 (E) such that E ⊆ K n0 .For every n ∈ N, let us define (4.17) Moreover, let us define (4.18) Applying [AS23, Theorem 2] to the convex set C , using U as open set, we get that there exists a closed strictly convex set C with nonempty interior and smooth boundary such that Notice that the pointed limits at infinity of C coincide with the pointed limits at infinity of C and the asymptotic cone of C coincides with the asymptotic cone of C .
Since I C (v) is not achieved on C , there is a pointed limit at infinity of C where there is an isoperimetric set with volume v and perimeter I C (v), see Lemma 2.3.Since also C has such pointed limit at infinity, we deduce, by Proposition 2.2, that I Assume by contradiction that there is a set E ⊆ C with |E| = v and Per C (E) = I C (v).Then the diameter of E is bounded above by L and therefore, by construction, there is a compact set K n0 such that E ⊆ K n0 .Then, by construction of C and by definition of To conclude, observe that since I C (v) is not achieved on C , it must be achieved on a pointed limit at infinity of C .And arguing as before we deduce I C (v) ≥ I C (v). □ 8 Here dist H denotes the Hausdorff distance defined by using the Euclidean distance.Since the function φ satisfies Eq. (1.19), we get that φ ′ < 0 and so φ is decreasing.Thus the quantity − h(φ(t)) is increasing and therefore φ ′ is increasing, which is equivalent to the convexity of φ.Also |φ ′ (t)| ≤ h(φ(t)) follows from Eq. (1.19).

Lemma 2. 1 .
Given d ≥ 2, let (C n ) n∈N and C be closed convex sets with nonempty interior in R d such that C n converges to C in the Hausdorff distance.Then C n converges to C in the pointed Gromov-Hausdorff sense (for any sequence of points ) in Proposition 2.19], see also [LRV22, Proposition 4.3] in the setting of convex bodies.The inequality I X∞ ≤ I X follows from the more general [BK23, Theorem 1.1], see also [LRV22, Theorem 6.3] in the setting of convex bodies.The last part of the statement follows from the rigidity statement in [Ant+23a, Item (3) of Theorem 1.2].□

d d− 1 X
under the assumption of existence of an isoperimetric set.We postpone to Remark 2.2 a comparison between Lemma 2.5 and the standard proof of the concavity of I d d−1

d d− 1
at v = v 0 and stays above (or equal) for all other values.The concavity of I

3. The construction 3 . 1 .
Presentation of the construction.Let d ≥ 2 be a natural number.Let Σ ⊆ R d be a subset such that (3.11) Σ is a closed convex cone with nonempty interior,

Proof.
It is proven repeating verbatim the proof of Proposition 3.3, up to replacing d with d − 1, | • | with H d , C with ∂C, Σ with ∂Σ.Moreover, one shall use Lemma 3.2 instead of Lemma 3.1 and Eq.(3.16) instead of Eq. (3.15).□

Proposition 4. 2 .
Given d ≥ 3, let C ⊆ R d be a convex set such that ∂C does not contain any line (or, equivalently, C does not split a line) and its asymptotic cone C ∞ is nondegenerate.Assume that, for some v > 0, any Borel set E ⊆ ∂C with H d−1 (E) = v satisfies Per ∂C (E) > I ∂C (v) (i.e., there is no isoperimetric set with volume v).There exists a strictly convex set C with smooth boundary so that distH (∂C \ B R , ∂ C \ B R ) → 0 as R → ∞ and any Borel set E ⊆ ∂ C with H d−1 (E) = v satisfies Per ∂ C (E) > I ∂ C (v) = I ∂C (v).Proof.It is proven repeating verbatim the proof of Proposition 4.1, up to replacingd with d − 1, | • | with H d−1 , C with ∂C , C with ∂ C appropriately.□5.Proof of the main resultsProof of Theorem 1.2.For d ≥ 3, let C ⊆ R d bethe convex set constructed in Proposition 3.3.Thus C does not have isoperimetric sets of volume 1.By Proposition 4.1 we can construct a strictly convex set C ⊆ R d with smooth boundary such that C does not have isoperimetric sets of volume 1.Notice that, thanks to the construction of C in Proposition 4.1, C has the same asymptotic cone and the same limits at infinity of C. Since C has nondegenerate asymptotic cone and all its limits at infinity are metric cones, see Section 3.2, we can apply Lemma 2.6 to C. Since C does not have isoperimetric sets of volume 1, we get that the threshold v 0 coming from Lemma 2.6 verifies v 0 > 0. After properly scaling in such a way v 0 = 1, the theorem is shown with C = C. □ Proof of Theorem 1.1.For d ≥ 3, by Proposition 3.4 there exists a convex set C ⊆ R d+1 such that ∂C ⊆ R d+1 does not have isoperimetric sets of volume 1.By Proposition 4.2 we can construct a strictly convex set C ⊆ R d+1 with smooth boundary ∂ C ⊆ R d+1 such that ∂ C does not have isoperimetric sets of volume 1.Notice that, thanks to the properties of C stated in Proposition 4.2, ∂ C has the same asymptotic cone and the same limits at infinity of ∂C.Observe that, by construction of C (see Section 3.1), all the limits at infinity of ∂C are cones, and ∂C has nondegenerate asymptotic cone.We are in position to apply Lemma 2.6 to ∂ C, and then obtain a threshold v 0 > 0. After scaling in such a way that v 0 = 1, we get the sought conclusion by taking M = ∂ C. □ of the Cauchy problem (local existence follows from [CL55, Chapter 1, Theorem 2.3], and global existence from the presence of the two mentioned barriers [CL55, Chapter 1, Theorem 4.1]).