Is a complete, reduced set necessarily of constant width?

TL;DR

This paper investigates whether complete and reduced convex bodies with respect to a gauge are necessarily of constant width, proving this under certain conditions and exploring implications for perfect norms.

Contribution

It establishes that complete and reduced convex bodies are of constant width in specific cases, advancing understanding of convex geometry and gauge bodies.

Findings

Complete and reduced bodies are of constant width if they are simplices or have a smooth extreme point.

New results on the properties of perfect norms are derived.

The paper extends known conditions under which convex bodies are 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 $K-K=\rho(C-C)$ for some $\rho>0$? 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.