Thrifty approximations of convex bodies by polytopes

Given a convex body C in R^d containing the origin in its interior and a real number tau > 1 we seek to construct a polytope P in C with as few vertices as possible such that C in tau P. Our construction is nearly optimal for a wide range of d and tau. In particular, we prove that if C=-C then for any 1>epsilon>0 and tau=1+epsilon one can choose P having roughly epsilon^{-d/2} vertices and for tau=sqrt{epsilon d} one can choose P having roughly d^{1/epsilon} vertices. Similarly, we prove that if C in R^d is a convex body such that -C in mu C for some mu > 1 then one can choose P having roughly ((mu+1)/(tau-1))^{d/2} vertices provided (tau-1)/(mu+1) << 1.