Isoperimetric inequalities in unbounded convex bodies

Leonardi, Gian Paolo; Ritoré, Manuel; Vernadakis, Efstratios

doi:10.48550/arxiv.1606.03906

Cited by 6 publications

(16 citation statements)

References 49 publications

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“…• The setting of RCD(0, N ) spaces (X, d, H N ) recovers in particular many of the results for Euclidean convex cones treated in [105] and for cones with nonnegative Ricci curvature considered in [94]. • The results of the present paper recover, in a much more general setting, many of the results proved in [83] for unbounded Euclidean convex bodies of uniform geometry.…”

Section: Isoperimetry and Lower Ricci Curvature Boundssupporting

confidence: 72%

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

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This paper studies sharp and rigid isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called N -dimensional RCD(K, N ) spaces (X, d, H N ). Thanks to these results, we determine the asymptotic isoperimetric behaviour for small volumes in great generality, and for large volumes when K = 0 under an additional noncollapsing assumption. Moreover, we obtain new stability results for isoperimetric regions along sequences of spaces with uniform lower Ricci curvature and lower volume bounds, almost regularity theorems formulated in terms of the isoperimetric profile, and enhanced consequences at the level of several functional inequalities.The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences.Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies. Contents 1. Introduction Isoperimetry and lower Ricci curvature bounds Main results Consequences and strategies of the proofs Sharp asymptotic behaviour for small and large volumes Comparison with the previous literature Acknowledgements 2. Preliminaries 2.1. Convergence and stability results 2.2. BV functions and sets of finite perimeter in metric measure spaces 2.3. Sobolev functions, Laplacians and vector fields in metric measure spaces 2.4. Geometric Analysis on RCD spaces 2.5. Localization of the Curvature-Dimension condition 3. The distance function from isoperimetric sets 4. Concavity properties of the isoperimetric profile function and consequences 4.1. Sharp concavity inequalities for the isoperimetric profile 4.2. Fine properties of the isoperimetric profile 4.3. Consequences 5. Stability of isoperimetric sets

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Section: Isoperimetry and Lower Ricci Curvature Boundssupporting

confidence: 72%

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

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“…The following lemma can be found in [21] for Carnot groups, and in the context of sub-Finsler nilpotent groups the proof can be done mutatis mutandis. The following Lemma can be found in [22]. Lemma 4.2.…”

Section: Existence Of Isoperimetric Regionsmentioning

confidence: 99%

“…In this paper we give a proof of existence of isoperimetric regions for any volume in a nilpotent group with a set of left-invariant vector fields X satisfying a Hörmander condition, that is, the Lie bracket generating condition, and an asymmetric leftinvariant norm • K , without the assumption of been equipped with a family of dilations (Theorem 4.3). We shall adapt the arguments of Galli and Ritoré [12] to prove existence of isoperimetric regions (see also [22]). As in [12], the main difficulty is to prove a Deformation Lemma (Lemma 3.6) which will allow us to increase the volume of any finite perimeter set while modifying the perimeter in a controlled way.…”

Section: Introductionmentioning

confidence: 99%

Existence of isoperimetric regions in sub-Finsler nilpotent groups

Pozuelo

2021

Preprint

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“…In Euclidean solid cones, the problem has been investigated in [76]. In more general convex bodies, it is treated in [57].…”

Section: Introductionmentioning

confidence: 99%

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

et al. 2022

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In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

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Isoperimetric inequalities in unbounded convex bodies

Cited by 6 publications

References 49 publications

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Existence of isoperimetric regions in sub-Finsler nilpotent groups

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

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