Approximations of convex bodies by polytopes and by projections of spectrahedra
Abstract
We prove that for any compact set B in Rd and for any epsilon >0 there is a finite subset X of B of |X|=dO(1/epsilon2) points such that the maximum absolute value of any linear function ell: Rd --> R on X approximates the maximum absolute value of ell on B within a factor of epsilon sqrtd. We also discuss approximations of convex bodies by projections of spectrahedra, that is, by projections of sections of the cone of positive semidefinite matrices by affine subspaces.
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