Steinhaus's geometric location problem for random samples in the plane

Abstract

LetwhereXi, 1 ≦i≦n, are i.i.d. and uniformly distributed in [0, 1]2. It is proved thatMn∽cn1–p/2a.s. for 1 ≦p<2. This result is motivated by recent developments in the theory of algorithms and the theory of subadditive processes as well as by a well-known problem of H. Steinhaus.