$\varepsilon$ -$\ell_p$不可压缩子集的等距降维
$\varepsilon$-isometric dimension reduction for incompressible subsets of $\ell_p$
摘要：
修复$p\in[1,\infty)$、$K\in(0,\infty)$和概率度量$μ$。我们证明，对于每个$n\in\mathbb{N}$，$\varepsilon\in(0,1)$和$x_1,\ldots,x_n\in L_p(μ)$与$\big\| \max_{i\in\{1,\ldots,n\}} |x_i| \big\|_{L_p(μ)} \leq K$，存在$d\leq \frac{32e^2 (2K)^{2p}\log n}{\varepsilon^2}$和矢量$y_1,\ldots, y_n \in \ell_p^d$，使$$\forall \ i,j\in\{1,\ldots,n\}, \qquad \|x_i-x_j\|^p_{L_p(μ)}- \varepsilon \leq \|y_i-y_j\|_{\ell_p^d}^p \leq \|x_i-x_j\|^p_{L_p(μ)}+\varepsilon.$$此外，参数意味着存在一个贪婪的算法，输出$\{y_i\}_{i=1}^n$后，收到$\{x_i\}_{i=1}^n$作为输入。证明依赖于Maurey的经验方法（1981）的去随机化版本，结合Ball（1990）的组合思想和$L_p(μ)$空间的经典因子分解理论。受上述嵌入的启发，我们引入了赋范空间$(E,\|\cdot\|_E)$的单位球${\bf B}_E$的$\varepsilon$ -等距降维的概念，并证明了对于$(E,\|\cdot\|_E)$的任意值，${\bf B}_{\ell_p}$不允许线性算子的$\varepsilon$ -等距降维.
英文摘要：
Fix $p\in[1,\infty)$, $K\in(0,\infty)$ and a probability measure $μ$. We prove that for every $n\in\mathbb{N}$, $\varepsilon\in(0,1)$ and $x_1,\ldots,x_n\in L_p(μ)$ with $\big\| \max_{i\in\{1,\ldots,n\}} |x_i| \big\|_{L_p(μ)} \leq K$, there exists $d\leq \frac{32e^2 (2K)^{2p}\log n}{\varepsilon^2}$ and vectors $y_1,\ldots, y_n \in \ell_p^d$ such that $$\forall \ i,j\in\{1,\ldots,n\}, \qquad \|x_i-x_j\|^p_{L_p(μ)}- \varepsilon \leq \|y_i-y_j\|_{\ell_p^d}^p \leq \|x_i-x_j\|^p_{L_p(μ)}+\varepsilon.$$ Moreover, the argument implies the existence of a greedy algorithm which outputs $\{y_i\}_{i=1}^n$ after receiving $\{x_i\}_{i=1}^n$ as input. The proof relies on a derandomized version of Maurey's empirical method (1981) combined with a combinatorial idea of Ball (1990) and classical factorization theory of $L_p(μ)$ spaces. Motivated by the above embedding, we introduce the notion of $\varepsilon$-isometric dimension reduction of the unit ball ${\bf B}_E$ of a normed space $(E,\|\cdot\|_E)$ and we prove that ${\bf B}_{\ell_p}$ does not admit $\varepsilon$-isometric dimension reduction by linear operators for any value of $p\neq2$.
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