加强Milman问题的一个反例
A counterexample to a strengthening of a question of Milman
摘要：
令$|\cdot|$是$\mathbb{R}^n$上的标准欧氏范数，令$X=(\mathbb{R}^n,\|\cdot\|)$是赋范空间。子空间$Y\subset X$是\emph{strongly $α$-Euclidean}，如果存在一个常数$t$，使得对于每个$y\in Y$都有$t|y|\leq\|y\|\leqαt|y|$，并且说它是\emph{strongly $α$-complemented}，如果$\|P_Y\|\leqα$，其中$P_Y$是从$X$到$Y$的正交投影，$\|P_Y\|$表示$P_Y$相对于$X$上的范数的算子范数。我们给出了一个例子，赋范空间$X$的任意高维，是强2欧几里德，但不包含2维子空间，这是强$(1+ε)$ -欧几里德和强$(1+ε)$ -补充，其中$ε>0$是一个绝对常数。这个例子与维塔利·米尔曼的一个老问题密切相关。
英文摘要：
Let $|\cdot|$ be the standard Euclidean norm on $\mathbb{R}^n$ and let $X=(\mathbb{R}^n,\|\cdot\|)$ be a normed space. A subspace $Y\subset X$ is \emph{strongly $α$-Euclidean} if there is a constant $t$ such that $t|y|\leq\|y\|\leqαt|y|$ for every $y\in Y$, and say that it is \emph{strongly $α$-complemented} if $\|P_Y\|\leqα$, where $P_Y$ is the orthogonal projection from $X$ to $Y$ and $\|P_Y\|$ denotes the operator norm of $P_Y$ with respect to the norm on $X$. We give an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is both strongly $(1+ε)$-Euclidean and strongly $(1+ε)$-complemented, where $ε>0$ is an absolute constant. This example is closely related to an old question of Vitali Milman.
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