On a relative isodiametric inequality for centrally symmetric, compact, convex surfaces

Abstract

We consider the problem of dividing a centrally symmetric, compact, convex surface into two regions in such a way that the intrinsic diameter of both regions is as small as possible. We discuss the best upper bound for the ratio between the area of the smallest region (relative area) and the maximal relative intrinsic diameter. We provide necessary and sufficient conditions for attaining the equality sign. As a consequence from these conditions, there are many surfaces for which the equality sign is never attained. We present a complete study of the special case of the cube surface obtaining the best possible upper bound for this surface.