On the vertex index of convex bodies

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We introduce the vertex index, vein ( K ) , of a given centrally symmetric convex body K ⊂ 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 ⊂ R d one has d 3 / 2 2 π e ovr ( K ) ⩽ vein ( K ) ⩽ C d 3 / 2 ln ( 2 d ) , where ovr ( K ) = inf ( vol ( E ) / vol ( K ) ) 1 / d 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 (respectively ellipsoid) is 2 d d . We prove this conjecture in dimensions two and three.