Is a complete, reduced set necessarily of constant width?

Abstract

Is it true that a convex body $K$ being complete and reduced with respect to some gauge body $C$ is necessarily of constant width, that is, satisfies $K-K=\ddot{I}(C-C)$ for some $\ddot{I}>0$? We prove this implication for several cases including the following: if $K$ is a simplex and or if $K$ possesses a smooth extreme point, then the implication holds. Moreover, we derive several new results on perfect norms.

12 pages