Finite coverings by translates of centrally symmetric convex domains

Abstract

Bambah and Rogers proved that the area of a convex domain in the plane which can be covered by n translates of a given centrally symmetric convex domain C is at most (n-1)h(C)+a(C), where h(C) denotes the area of the largest hexagon contained in C and a(C) stands for the area of C. An improvement with a term of magnitude √n is given here. Our estimate implies that if C is not a parallelogram, then any covering of any convex domain by at least 26 translates of C is less economic than the thinnest covering of the whole plane by translates of C. © 1987 Springer-Verlag New York Inc.