Convex bodies with few faces

It is proved that if $u_1,\ldots, u_n$ are vectors in ${\Bbb R}^k, k\le n, 1 \le p < \infty$ and $r = ({1\over k} \sum ^n_1 |u_i|^p)^{1\over p}$ then the volume of the symmetric convex body whose boundary functionals are $\pm u_1,\ldots, \pm u_n$, is bounded from below as $|\{ x\in {\Bbb R}^k\colon \ |\langle x,u_i\rangle | \le 1 \ \hbox{for every} \ i\}|^{1\over k} \ge {1\over \sqrtρr}.$ An application to number theory is stated.

No reviews yet.

No comments yet.