Failure of Convex-Hull Bounds under Log-Convex Tails

Fix $0<r<1$, and let $X_1,X_2,\dots$ be independent symmetric Weibull$(r)$ random variables, that is, $$\mathsf{\text{P}}(\mathrm{\mid }{X}_i\mathrm{\mid }>t)=e^{-t^r},t\ge 0.$$ We prove that there is no constant $C_r$, depending only on $r$, with the following universal property: for every finite set $T\subset {\mathbb{R}}^N$ there exists a sequence $(y_k{)}_{k\ge 1}\subset {\mathbb{R}}^N$ such that $$T-T\subset conv\{y_k:k\ge 1\},\mathrm{\parallel }{X}_{y_k}{\mathrm{\parallel }}_{L_{\log (k+2)}}\le C_r\text{bx}(T)(k\ge 1),$$ where $X_t={\sum }_i{t}_i{X}_i$ and $\text{bx}(T)=\mathsf{\text{E}}{\sup }_{t\in T}{X}_t$. This gives a negative answer to a question of Latała 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.