Failure of Convex-Hull Bounds under Log-Convex Tails

Xuanang Hu; Hanchao Wang

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arXiv · 2607.00538

Failure of Convex-Hull Bounds under Log-Convex Tails

Xuanang Hu·Hanchao Wang

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Abstract

Fix $0 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 validity of convex-hull bounds for canonical Weibull processes. In fact, the failure persists even when the auxiliary vectors appearing in the convex hull are allowed to be arbitrary.

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Xuanang Hu, Hanchao Wang. 2026-07-01. Failure of Convex-Hull Bounds under Log-Convex Tails. https://arxiv.org/abs/2607.00538

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