一个完备的约化集是否一定具有恒定的宽度？
Is a complete, reduced set necessarily of constant width?
摘要：
完备的凸体$K$相对于某个规范体$C$约化了，它是否必然具有恒定的宽度，即对于某个$ρ>0$满足$K-K=ρ(C-C)$？我们证明了几种情况下，包括以下情况：如果$K$是一个单纯形和或如果$K$具有一个光滑的极值点，则蕴涵保持。此外，我们还得到了一些关于完全范数的新结果。
英文摘要：
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 $K-K=ρ(C-C)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
