Vertex Index of Symmetric Convex Bodies
Thursday, May 7, 2015 - 2:00pm - 3:00pm
We discuss several results on the vertex index of a given $d$-dimensional centrally symmetric convex body, which, in a sense, measures how well the body 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 Karoly Bezdek, 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 estimates of this index and relate the lower bound with the outer volume ratio. We also discuss sharpness of the bounds, providing examples. The talk is based on joint works with K.Bezdek and E.D.Gluskin.