{"paper":{"title":"Do Minkowski averages get progressively more convex?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OC"],"primary_cat":"math.MG","authors_text":"Arnaud Marsiglietti, Artem Zvavitch, Matthieu Fradelizi, Mokshay Madiman","submitted_at":"2015-12-05T14:11:50Z","abstract_excerpt":"Let us define, for a compact set $A \\subset \\mathbb{R}^n$, the Minkowski averages of $A$: $$ A(k) = \\left\\{\\frac{a_1+\\cdots +a_k}{k} : a_1, \\ldots, a_k\\in A\\right\\}=\\frac{1}{k}\\Big(\\underset{k\\ {\\rm times}}{\\underbrace{A + \\cdots + A}}\\Big). $$ We study the monotonicity of the convergence of $A(k)$ towards the convex hull of $A$, when considering the Hausdorff distance, the volume deficit and a non-convexity index of Schneider as measures of convergence. For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03718","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":""},"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"}