On the vertex index of convex bodies

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K ⊂ Rd, 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 2d 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 ⊂ Rd one has d3/2 √ 2πe ovr(K) ≤ vein(K) ≤ C d ln(2d), where ovr(K) = inf (vol (E)/ vol (K)) is the outer volume ratio of K with the infimum taken over all ellipsoids E ⊃ 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 (resp., ellipsoid) is 2d √ d. We prove this conjecture in dimensions two and three. ∗