On the vertex index of convex bodies

We introduce the vertex index, vein(K)vein(K), of a given centrally symmetric convex body K⊂RdK⊂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 d22d 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⊂RdK⊂Rd one hasd3/22πeovr(K)⩽vein(K)⩽Cd3/2ln(2d), where ovr(K)=inf(vol(E)/vol(K))1/dovr(K)=inf(vol(E)/vol(K))1/d is the outer volume ratio of K with the infimum taken over all ellipsoids E⊃KE⊃K and with vol(⋅)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)⩽64⩽vein(K)⩽6 with equalities for parallelograms and affine regular convex hexagons, and in the 3-dimensional case we show that 6⩽vein(K)6⩽vein(K) with equality for octahedra. We conjecture that the vertex index of a d  -dimensional Euclidean ball (respectively ellipsoid) is 2dd. We prove this conjecture in dimensions two and three.