Covering the boundary of a convex body with its smaller homothetic copies

摘要

For each positive integer m and any convex body K, denote by gamma(m)(K) the smallest positive number gamma so that the boundary of K can be covered by m translates of gamma K. It is proved that, for each positive integer m, gamma(m)(K) is Lipschitz continuous on the space of affine equivalence classes of n-dimensional convex bodies endowed with the Banach-Mazur metric. Exact values of gamma(m)(K) for particular choices of planar convex bodies K and positive integers m are also obtained. Moreover, a general way to estimate gamma(m)(K) for centrally symmetric convex bodies is presented. (C) 2018 Elsevier B.V. All rights reserved.