Is a complete, reduced set necessarily of constant width?

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 $K-K=ρ(C-C)$ for some $ρ>0$? 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.

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