{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KIE56TZYAOX3DZQXAK6JZHOEKU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f6ee7138c4f95b635d77590891cf3069c480008eb6656a3771fd85a519a54ace","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-02T19:47:53Z","title_canon_sha256":"59235e97f4a370b22ee3f80b8193b6d313e2048a7342ff814ffe80908f3372f2"},"schema_version":"1.0","source":{"id":"1306.0246","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.0246","created_at":"2026-05-18T03:21:55Z"},{"alias_kind":"arxiv_version","alias_value":"1306.0246v1","created_at":"2026-05-18T03:21:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0246","created_at":"2026-05-18T03:21:55Z"},{"alias_kind":"pith_short_12","alias_value":"KIE56TZYAOX3","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KIE56TZYAOX3DZQX","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KIE56TZY","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:5c13d404f980649768e635a71c8d6be1e431b97e2e45c967ba94e0776837bd20","target":"graph","created_at":"2026-05-18T03:21:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study some geometric properties of the $L_q$-centroid bodies $Z_q(\\mu)$ of an isotropic log-concave measure $\\mu $ on ${\\mathbb R}^n$. For any $2\\ls q\\ls\\sqrt{n}$ and for $\\varepsilon \\in (\\varepsilon_0(q,n),1)$ we determine the inradius of a random $(1-\\varepsilon)n$-dimensional projection of $Z_q(\\mu)$ up to a constant depending polynomially on $\\varepsilon $. Using this fact we obtain estimates for the covering numbers $N(\\sqrt{\\smash[b]{q}}B_2^n,tZ_q(\\mu))$, $t\\gr 1$, thus showing that $Z_q(\\mu)$ is a $\\beta $-regular convex body. As a consequence, we also get an upper bound for $M(Z_q(","authors_text":"Antonis Tsolomitis, Apostolos Giannopoulos, Beatrice-Helen Vritsiou, Pantelis Stavrakakis","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-02T19:47:53Z","title":"Geometry of the $L_q$-centroid bodies of an isotropic log-concave measure"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0246","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6c808b21c16e1970d727f408159e98c0841f410fa065d9859279273221839718","target":"record","created_at":"2026-05-18T03:21:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f6ee7138c4f95b635d77590891cf3069c480008eb6656a3771fd85a519a54ace","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-06-02T19:47:53Z","title_canon_sha256":"59235e97f4a370b22ee3f80b8193b6d313e2048a7342ff814ffe80908f3372f2"},"schema_version":"1.0","source":{"id":"1306.0246","kind":"arxiv","version":1}},"canonical_sha256":"5209df4f3803afb1e61702bc9c9dc455249b3290add00458c156fba3bca072f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5209df4f3803afb1e61702bc9c9dc455249b3290add00458c156fba3bca072f8","first_computed_at":"2026-05-18T03:21:55.710172Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:55.710172Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WvkBVus8GHlpzVulZr/uE/e4G1jeJCX9N8mXhVyNSLGfrTAIkeTaD8hZRjdpTMmF7Es+5XOo01mKCT4gX1KPCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:55.710660Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.0246","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c808b21c16e1970d727f408159e98c0841f410fa065d9859279273221839718","sha256:5c13d404f980649768e635a71c8d6be1e431b97e2e45c967ba94e0776837bd20"],"state_sha256":"0c98bd228e6a3dec6f09f27b132abb443672c077549e1c7167333fd6dcbff069"}