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Blaschke 1917 \\cite{Bla} proved that $Q^4_T\\leq Q^4_H\\leq Q^4_D$, where $D$ is a disk and $T$ a triangle. In the present paper we prove $Q^5_T\\leq Q^5_H\\leq Q^5_D$. One of the main ingredients of our approach is a new formula for $Q^n_H$ which permits to prove that Steiner symmetrization does not decrease $Q^5_H$, and that shaking does not increases it (this is the meth"},"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":"1511.03658","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-11T20:59:02Z","cross_cats_sorted":[],"title_canon_sha256":"75aabc902dc4c414733a9a3d5b9d6458db80d0658117f879d9ba6dfa7fb4ad15","abstract_canon_sha256":"5e4bedd0975466b92d14ba6fd352234a2ac4b42bafa03c258ae1453a8c201962"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:07.082180Z","signature_b64":"ShuIkLCVXgTmbyeN5ba3B0vLXj4a3nfxEk6Tb4ILgV1OSuhzrdGBvPzFaAEMVUJOzeyJZLcPDrGzxUkeYfR9Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5dadf2446cbed6bccf7bb4d5cc904481317a9ca70494b39d22cc55f8393af25b","last_reissued_at":"2026-05-18T01:27:07.081642Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:07.081642Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Around Sylvester's question in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jean-Fran\\c{c}ois Marckert","submitted_at":"2015-11-11T20:59:02Z","abstract_excerpt":"Pick $n$ points $Z_0,...,Z_{n-1}$ uniformly and independently at random in a compact convex set $H$ with non empty interior of the plane, and let $Q^n_H$ be the probability that the $Z_i$'s are the vertices of a convex polygon. 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