Covering the Boundary of a Convex Body with Its Smaller Homothetic Copies

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