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

For every $2<p<\mathrm{\infty }$ and every integer $r\ge 2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $\mathrm{\mid }T\mathrm{\mid }\le 2^{C{r}^2}$, ${\gamma }_2(T)\le Cr$, and ${\gamma }_2(\text{conv}T)\ge c{r}^{3\mathrm{/}2-1\mathrm{/}p}$. 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 ${\mathrm{\ell }}_p$. 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 ${\mathrm{\ell }}_p^r({\mathrm{\ell }}_2^{2^{128r}})$ for $2<p\le \mathrm{\infty }$ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.