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

Abstract

For each positive integer m and any convex body K, denote by γm(K) the smallest positive number γ so that the boundary of K can be covered by m translates of γK. It is proved that, for each positive integer m, γm(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 γm(K) for particular choices of planar convex bodies K and positive integers m are also obtained. Moreover, a general way to estimate γm(K) for centrally symmetric convex bodies is presented.