{"paper":{"title":"Failure of Convex-Hull Bounds under Log-Convex Tails","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Hanchao Wang, Xuanang Hu","submitted_at":"2026-07-01T07:28:32Z","abstract_excerpt":"Fix $0<r<1$, and let $X_1,X_2,\\dots$ be independent symmetric Weibull$(r)$ random variables, that is, \\[ \\textsf{P}(|X_i|>t)=e^{-t^r},\\qquad t\\ge 0. \\] We prove that there is no constant $C_r$, depending only on $r$, with the following universal property: for every finite set $T\\subset \\R^N$ there exists a sequence $(y_k)_{k\\ge 1}\\subset \\R^N$ such that \\[ T-T\\subset conv\\{y_k:k\\ge 1\\}, \\qquad \\|X_{y_k}\\|_{L_{\\log(k+2)}}\\le C_r\\,\\bx(T) \\quad (k\\ge 1), \\] where $X_t=\\sum_i t_i X_i$ and $\\bx(T)=\\textsf{E}\\sup_{t\\in T}X_t$. This gives a negative answer to a question of Lata{\\l}a concerning the va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00538","kind":"arxiv","version":1},"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/2607.00538/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"}