AI 中文总结
该注记研究勒贝格通用覆盖问题，证明三维外接直径1球的正十二面体非通用覆盖，改进三维通用覆盖最小体积界，给出$d$维外接单位球的最大面数中心对称通用覆盖多面体族，推广并改进了相关界。
AI 中文摘要
通用覆盖是指经过旋转、反射和平移后能覆盖所有直径为1的集合的凸集。首先，我们证明在三维欧氏空间$\boldsymbol{\text{R}^3}$中，外接直径为1的球的正十二面体不是通用覆盖，回答了查克里安（Chakerian）提出的问题[K]。其次，我们将三维欧氏空间$\boldsymbol{\text{R}^3}$中通用覆盖的最小体积界改进为$(0.545193,0.655984)$。第三，我们给出$d$维欧氏空间$\boldsymbol{\text{R}^d}$中外接单位球、具有$d(d+1)$个面的中心对称多面体的无限族，这类多面体是通用覆盖，且面数达到最大可能值，证明了马基耶夫（Makeev）提出的猜想[M2]。最后，针对$d$维欧氏空间$\boldsymbol{\text{R}^d}$中外接单位球且非通用覆盖的中心对称多面体的最少面数问题，我们推广了马基耶夫的下界[M1]并改进了他的上界[M3]。
英文摘要
A universal cover is a convex set that covers all sets of diameter 1 after some rotation, reflection, and translation. First, we show that a regular dodecahedron that circumscribes a ball of diameter 1 is not a universal cover in $\mathbb R^3$, answering a question of Chakerian [K]. Second, we improve the bounds on the minimum volume of a universal cover in $\mathbb R^3$ to $(0.545193,0.655984)$. Third, we give an infinite family of centrally symmetric polytopes in $\mathbb R^d$ with $d(d+1)$ facets that circumscribe the unit ball and are universal covers. This is the maximum possible number and proves a conjecture of Makeev [M2]. Finally, on the least number of facets of a centrally symmetric polytope in $\mathbb R^d$ that circumscribe the unit ball and is not a universal cover, we generalize Makeev's lower bound [M1] and improve his upper bound [M3].
Comments13 pages, 2 figures
