关于John-Löwner椭球的体积
On the volume of the John-Löwner ellipsoid
摘要：
我们发现一个最佳的上界的体积的约翰椭球的一个$k$维部分的$n$维立方体，和一个最佳的下界的体积的Löwner椭球的投影的$n$维交叉多面体到一个$k$维子空间。利用这些结果，我们给出了超立方体的$k$维截面的体积的Ball上界和$k$维交叉多面体在$k$维子空间上的投影的体积的Barthe下界的新证明.我们在这些不平等中解决平等问题。此外，我们描述了所有可能的矢量在$\R^n,$，其坐标是平方长度的投影的标准基础在$\R^n$到一个$k$维子空间。
英文摘要：
We find an optimal upper bound on the volume of the John ellipsoid of a $k$-dimensional section of the $n$-dimensional cube, and an optimal lower bound on the volume of the Löwner ellipsoid of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We use these results to give a new proof of Ball's upper bound on the volume of a $k$-dimensional section of the hypercube, and of Barthe's lower bound on the volume of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We settle equality cases in these inequalities. Also, we describe all possible vectors in $\R^n,$ whose coordinates are the squared lengths of a projection of the standard basis in $\R^n$ onto a $k$-dimensional subspace.
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