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 $K-K=\rho(C-C)$ 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. 
Ren\'e Brandenberg, Bernardo Gonz\'alez Merino, Thomas Jahn, and Horst   Martini
math.MG