English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove the sharp quantitative stability for a wide class of weightedisoperimetric inequalities. More precisely, we consider isoperimetricinequalities in convex cones with homogeneous weights.  Inspired by the proof of such isoperimetric inequalities through the ABPmethod, we construct a new convex coupling (i.e., a map that is the gradient ofa convex function) between a generic set $E$ and the minimizer of theinequality (as in Gromov's proof of the isoperimetric inequality). Even if thismap does not come from optimal transport, and even if there is a weight in theinequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if$E$ is almost optimal for the inequality then it is quantitatively close to aminimizer up to translations. Then, a delicate analysis is necessary to ruleout the possibility of translations.  As a step of our proof, we establish a sharp regularity result for restrictedconvex envelopes of a function that might be of independent interest.

Sharp quantitative stability for isoperimetric inequalities with homogeneous weights