Convex bodies with few faces

Abstract

It is proved that if $u_1,\dots ,u_n$ are vectors in ${\mathbb{R}}^k,k\le n,1\le p<\infty$ and $$r=(\frac{1}{k}\sum _1^n|u_i{|}^p{)}^{\frac{1}{p}}$$ then the volume of the symmetric convex body whose boundary functionals are $\pm u_1,\dots ,\pm u_n$, is bounded from below as $$|\{x\in {\mathbb{R}}^k:|⟨x,u_i⟩|\le 1for\; everyi\}{|}^{\frac{1}{k}}\ge \frac{1}{\sqrt{\rho }r}.$$ An application to number theory is stated.