On the three-dimensional Blaschke-Lebesgue problem

Abstract

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n=3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant, and therefore are either spherical caps or pieces of tubes (canal surfaces).