坐标子空间上的低$M^*$ -估计
Low $M^*$-estimates on coordinate subspaces
摘要：
设$K$是${\mathbf R}^n$中的对称凸体。众所周知，对于每个$θ\in (0,1)$，存在具有${\rm dim}F= [(1-θ)n]$的${\mathbf R}^n$的子空间$F$，使得$${\mathcal P}_F(K)\supseteq \frac{c\sqrt{θ}} {M_K}D_n\cap F,\leqno (\ast )$$，其中${\mathcal P}_F$表示到$F$上的正交投影。考虑${\mathbf R}^n$中的固定坐标系。本文研究了当人们只能在坐标子空间${\mathbf R}^{σ},\; σ\subseteq\{1,\ldots,n\}$中选择$F$时，用$|σ|=[(1-θ)n]$是否能得到（$\ast $）的一个模拟.我们证明了（$\ast $）的几个"坐标形式”，它们是关于$K$的cotype-2常数、体积比和其它参数的.我们的估计的基本来源是一个精确的坐标模拟（$\ast $）在椭球的情况下。应用程序的凸体内部的格点的数量的计算被认为是整个文件。
英文摘要：
Let $K$ be a symmetric convex body in ${\mathbf R}^n$. It is well-known that for every $θ\in (0,1)$ there exists a subspace $F$ of ${\mathbf R}^n$ with ${\rm dim}F= [(1-θ)n]$ such that $${\mathcal P}_F(K)\supseteq \frac{c\sqrt{θ}} {M_K}D_n\cap F,\leqno (\ast )$$ where ${\mathcal P}_F$ denotes the orthogonal projection onto $F$. Consider a fixed coordinate system in ${\mathbf R}^n$. We study the question whether an analogue of ($\ast $) can be obtained when one is restricted to choose $F$ among the coordinate subspaces ${\mathbf R}^{σ},\; σ\subseteq\{1,\ldots,n\}$, with $|σ|=[(1-θ)n]$. We prove several ``coordinate versions" of ($\ast $) in terms of the cotype-2 constant, of the volume ratio and other parameters of $K$. The basic source of our estimates is an exact coordinate analogue of ($\ast $) in the ellipsoidal case. Applications to the computation of the number of lattice points inside a convex body are considered throughout the paper.
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