On the vertex index of convex bodies

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K subset of R-d, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2(d) smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K subset of R-d one has [GRAPHICS] where ovr(K) = inf(vol(epsilon)/vol(K)()1/d) is the outer volume ratio of K with the infimum taken over all ellipsoids epsilon superset of K and with vol(.) denoting the volume. Also, we provide sharp estimates in dimensions 2 and 3. Namely, in the planar case we prove that 4 <= vein(K) <= 6 with equalities for parallelograms and affine regular convex hexagons, and in the 3-dimensional case we show that 6 <= vein(K) with equality for octahedra. We conjecture that the vertex index of a d-dimensional Euclidean ball (respectively ellipsoid) is 2d root d. We prove this conjecture in dimensions two and three. (c) 2007 Elsevier Inc. All rights reserved.