Convex-Hull Instability of the ${\gamma }_2$-Functional in $L^p$

Abstract

For every $2<p<\infty$ and every integer $r\ge 2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $|T|\le 2^{C{r}^2}$, ${\gamma }_2(T)\le Cr$, and $γ_2(\conv T)\ge c r^{3/2-1/p}$. 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 $2$.

arXiv commentCurrent frontier generative-AI models can produce essentially similar constructions when presented with the underlying problem. The authors do not plan to submit this work to a journal