标题： 一个完全的、约简的集合是否一定是等宽的？ 显示英文标题

标题： Is a complete, reduced set necessarily of constant width?

摘要： 凸体$K$在相对于某个度量体$C$完全且约简的情况下，是否一定是等宽的，即满足$K-K=\rho(C-C)$对某个$\rho>0$？ 我们证明了这一推论在几种情况下成立，包括以下情况：如果$K$是单形，或者如果$K$拥有一个光滑的极点，则该推论成立。 此外，我们得出了关于完美范数的几个新结果。 显示英文摘要

摘要： 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.