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For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1512.03718","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-12-05T14:11:50Z","cross_cats_sorted":["math.FA","math.OC"],"title_canon_sha256":"7332ab348bf6630a710a4552a910085cdfaf8bb43968ca37ac86ce8d56e42e6b","abstract_canon_sha256":"0da1eacc1958e57cf913d6e159cb8abfc9c90f320bd6bb0db54111ec4e784f44"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:11.071607Z","signature_b64":"2sI4sFH9PxAPnUNyZBu1C/kYKmoqEbjd1gf0QmBPBgguHBUfgAyuq/+iYB2pk8NowFSVF4Oluge2Vok+SsH9Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1f521a59279bab2ca204fc7faa3b14dbf3919d6657cf86f4fd936b387ed0031","last_reissued_at":"2026-05-18T01:21:11.071085Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:11.071085Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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