We prove that for any compact set B in R^d and for any epsilon >0 there is a
finite subset X of B of |X|=d^{O(1/epsilon^2)} points such that the maximum
absolute value of any linear function ell: R^d --> R on X approximates the
maximum absolute value of ell on B within a factor of epsilon sqrt{d}. 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.Comment: 13 pages, some improvements, acknowledgment adde