{"paper":{"title":"Sharp quantitative stability of the planar Brunn-Minkowski inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Hunter Spink, Marius Tiba, Peter van Hintum","submitted_at":"2019-11-27T04:17:43Z","abstract_excerpt":"We prove a sharp stability result for the Brunn-Minkowski inequality for $A,B\\subset\\mathbb{R}^2$. Assuming that the Brunn-Minkowski deficit $\\delta=|A+B|^{\\frac{1}{2}}/(|A|^\\frac12+|B|^\\frac12)-1$ is sufficiently small in terms of $t=|A|^{\\frac{1}{2}}/(|A|^{\\frac{1}{2}}+|B|^{\\frac{1}{2}})$, there exist homothetic convex sets $K_A \\supset A$ and $K_B\\supset B$ such that $\\frac{|K_A\\setminus A|}{|A|}+\\frac{|K_B\\setminus B|}{|B|} \\le C t^{-\\frac{1}{2}}\\delta^{\\frac{1}{2}}$. The key ingredient is to show for every $\\epsilon>0$, if $\\delta$ is sufficiently small then $|co(A+B)\\setminus (A+B)|\\le ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.11945","kind":"arxiv","version":1},"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/1911.11945/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"}