Intersections of Convex Bodies with Their Translates

Abstract

It has been shown by Fujiwara [4] and Bol [2] that if K is a planar convex body which is not a disk, then it is possible to find a congruent copy K′ of K such that K and K′ have more than two points in common on their boundaries. This result was used by Yanagihara [9] to show that if K is a 3-dimensional convex body with the property that for any congruent copy K′ of K, the boundaries of K and K′ intersect in a planar curve (assuming they do, in fact, meet but do not coincide), then K is a ball.