{"paper":{"title":"Strong stability of convexity with respect to the perimeter","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Alessio Figalli, Yi Ru-Ya Zhang","submitted_at":"2023-07-04T10:28:33Z","abstract_excerpt":"Let $E\\subset \\mathbb R^n$, $n\\ge 2$, be a set of finite perimeter with $|E|=|B|$, where $B$ denotes the unit ball. When $n=2$, since convexification decreases perimeter (in the class of open connected sets), it is easy to prove the existence of a convex set $F$, with $|E|=|F|$, such that $$\n  P(E) - P(F) \\ge c\\,|E\\Delta F|, \\qquad c>0. $$ Here we prove that, when $n\\ge 3$, there exists a convex set $F$, with $|E|=|F|$, such that $$\n  P(E) - P(F) \\ge c(n) \\,f\\big(|E\\Delta F|\\big), \\qquad c(n)>0,\\qquad f(t)=\\frac{t}{|\\log t|} \\text{ for }t \\ll 1. $$ Moreover, one can choose $F$ to be a small $C"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.01633","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.01633/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}