{"paper":{"title":"On the volume of convolution bodies in the plane","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"J. Haddad","submitted_at":"2024-04-30T21:35:25Z","abstract_excerpt":"For every convex body $K \\subset \\mathbb R^n$ and $\\delta \\in (0,1)$, the $\\delta$-convolution body of $K$ is the set of $x \\in \\mathbb R^n$ for which $\\left|K \\cap (K+x)\\right|_n \\geq \\delta \\left|K\\right|_n$.\n  We show that for $n=2$ and any $\\delta \\in (0,1)$, ellipsoids do not maximize the volume of the $\\delta$-convolution body of $K$, when $K$ runs over all convex bodies of a fixed volume.\n  This behavior is somehow unexpected and contradicts the limit case $\\delta \\to 1^-$, which is governed by the Petty projection inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.00212","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/2405.00212/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"}