标题： Macbeath定理在有限维赋范空间中的变体 显示英文标题

标题： Variants of a theorem of Macbeath in finite dimensional normed spaces

摘要： 一个经典的Macbeath定理指出，对于任何整数$d \geq 2$，$n \geq d+1$，$d$-维欧几里得球在体积差的意义下最难用内接凸多面体近似，这些凸多面体有$n$个顶点。 在本文中，我们研究这个问题的赋范变体：我们旨在找到Busemann体积、Holmes-Thompson体积、Gromov的质量和Gromov的质量$^*$的最大体积凸多面体的极值，该凸多面体有$n$个顶点，内接于一个$d$-维赋范空间的单位球中。 显示英文摘要

摘要： A classical theorem of Macbeath states that for any integers $d \geq 2$, $n \geq d+1$, $d$-dimensional Euclidean balls are hardest to approximate, in terms of volume difference, by inscribed convex polytopes with $n$ vertices. In this paper we investigate normed variants of this problem: we intend to find the extremal values of the Busemann volume, Holmes-Thompson volume, Gromov's mass and Gromov's mass$^*$ of a largest volume convex polytope with $n$ vertices, inscribed in the unit ball of a $d$-dimensional normed space.