Approximations of convex bodies by polytopes and by projections of spectrahedra

Details Approximations of convex bodies by polytopes and by projections of spectrahedra Approximations of convex bodies by polytopes and by projections of spectrahedra

2012-04-02

arxiv.org/abs/1204.0471v2

Mathematics

Metric Geometry

13 pages, some improvements, acknowledgment added

Scientific paper

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.

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