  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.
