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.