{"paper":{"title":"Volume of the Minkowski sums of star-shaped sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Artem Zvavitch, Matthieu Fradelizi, Zsolt L\\'angi","submitted_at":"2019-10-14T13:59:29Z","abstract_excerpt":"For a compact set $A \\subset {\\mathbb R}^d$ and an integer $k\\ge 1$, let us denote by $$ A[k] = \\left\\{a_1+\\cdots +a_k: a_1, \\ldots, a_k\\in A\\right\\}=\\sum_{i=1}^k A$$ the Minkowski sum of $k$ copies of $A$. A theorem of Shapley, Folkmann and Starr (1969) states that $\\frac{1}{k}A[k]$ converges to the convex hull of $A$ in Hausdorff distance as $k$ tends to infinity. Bobkov, Madiman and Wang (2011) conjectured that the volume of $\\frac{1}{k}A[k]$ is non-decreasing in $k$, or in other words, in terms of the volume deficit between the convex hull of $A$ and $\\frac{1}{k}A[k]$, this convergence is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.06146","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/1910.06146/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"}