Convex-Hull Instability of the $\gamma_2$-Functional in $L^p$

Abstract

For every $2<p<\infty$ and every integer $r\ge2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $|T|\le2^{Cr^2}$, $\gamma_2(T)\le Cr$, and $\gamma_2(\conv T)\ge c r^{3/2-1/p}$. Consequently, for every fixed $p>2$, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces $L^p[0,1]$ and $\ell_p$. The lower bound follows from a multilevel product principle applied to a scale-separated product of Euclidean simplices. The same construction gives the sharp convexification profile of $\ell_p^r(\ell_2^{2^{128r}})$ for $2<p\le\infty$ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.