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   $$|\{ x\in {\Bbb R}^k\colon \ |\langle x,u_i\rangle | \le 1 \ \hbox{for every} \ i\}|^{1\over k} \ge {1\over \sqrt{\rho}r}.$$   An application to number theory is stated. 
Keith Ball (Texas A&M University) and Alain Pajor (Paris VII)