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

Helen W. J. Zhang, Chengdong Zhao

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}…