On the asymmetry constant of a body with few vertices

In this note we show that a non-degenerated polytope in IRn with n+k, 1 ≤ k < n, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes. 0 Introduction and notations The canonical Euclidean inner product in IR is denoted by 〈·, ·〉, the norm in `p is denoted by ‖ · ‖p, 1 ≤ p ≤ ∞. By a convex body K ⊂ IR we shall always mean a compact convex set with the non-empty interior. By a non-degenerated polytope we mean a convex polytope with the nonempty interior. Given convex bodies K, L in IR, we define the geometric distance by d̃(K,L) = inf{αβ | α > 0, β > 0, (1/β)L ⊂ K ⊂ αL}. Denote by C the set of all centrally symmetric with respect to the origin convex bodies in IR. For a convex body K in IR we define the asymmetry constant δ(K) of a convex body K as follows δ(K) := d̃(K, C) = inf { d̃(K − a,B) | a ∈ IR, B ∈ C } . ∗This author was supported by a Lady Davis Fellowship