Convex bodies with few faces

Abstract:

It is proved that it ${u_1}, \ldots ,{u_n}$ are vectors in ${{\mathbf {R}}^k},k \leq n,1 \leq p < \infty$ and \[ r = {\left ( {\frac {1}{k}{{\sum \limits _1^n {\left | {{u_i}} \right |} }^p}} \right )^{1/p}}\] then the volume of the symmetric convex body whose boundary functionals are $\pm {u_1}, \ldots , \pm {u_n}$, is bounded from below as \[ {\left | {\left \{ {x \in {{\mathbf {R}}^k}:\left | {\left \langle {x,{u_i}} \right \rangle } \right | \leq 1{\text { for every }}i} \right \}} \right |^{1/k}} \geq 1/\sqrt \rho r.\] An application to number theory is stated.