This is an announcement for the paper "Approximations of convex bodies by polytopes and by projections of spectrahedra" by Alexander Barvinok.

Abstract: 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 prove that for any finite set B in Z^d and for any positive integer k there is a convex set C in R^d containing B such that C is an affine image of a section of the cone of rxr positive semidefinite matrices for r=d^{O(k)} and such that for any linear function ell: R^d --> R with integer coefficients the maximum absolute value of ell on B and the maximum absolute value of ell on C coincide provided the former does not exceed k.

Archive classification: math.MG math.FA math.OC

Mathematics Subject Classification: 52A20, 52A27, 52A21, 52B55, 90C22

Remarks: 11 pages

Submitted from: barvinok@umich.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1204.0471

or

http://arXiv.org/abs/1204.0471