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2022-01-12

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cc-by-nc-nd (c) American Mathematical Society (AMS), 2022
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/194107

Sharp quantitative stability for isoperimetric inequalities with homogeneous weights

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https://doi.org/10.1090/tran/8525

Abstract

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method (see [CRS16]), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of [FMP10] and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.

Subject

Varietats (Matemàtica), Optimització matemàtica, Teoria de la mesura geomètrica

Subject (English)

Manifolds (Mathematics), Mathematical optimization, Geometric measure theory

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Articles publicats en revistes (Matemàtiques i Informàtica)

Citation

CINTI, Eleonora, et al. Sharp quantitative stability for isoperimetric inequalities with homogeneous weights. Transactions of the American Mathematical Society. 2022. Vol. 375, num. 1509-1555. ISSN 0002-9947. [consulted: 16 of August of 2026]. Available at: https://hdl.handle.net/2445/194107

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