CONTACT POINTS OF CONVEX-BODIES

Let B be a convex body in R-n and let E be an ellipsoid of minimal vol ume containing B. By contact points of B we mean the points of the int ersection between the boundaries of B and E. By a result of P. Gruber, a generic convex body in R-n has (n + 3). n/2 contact points. We prov e that for every epsilon > 0 and for every convex body B subset of R-n there exists a convex body K having m less than or equal to C(epsilon ). n log(3) n contact points whose Banach-Mazur distance to B is less than 1 + epsilon. We prove also that for every t > 1 there exists a co nvex symmetric body Gamma subset of R-n so that every convex body D su bset of R-n whose Banach-Mazur distance to Gamma is less than t has at least (1 + c(0)/t(2)). n contact points for some absolute constant co . We apply these results to obtain new factorizations of Dvoretzky-Rog ers type and to estimate the size of almost orthogonal submatrices of an orthogonal matrix.