论文标题
几乎最小的正交预测
Almost minimal orthogonal projections
论文作者
论文摘要
有限维度的Banach Space $ e \ subset \ ell_ \ infty $的投影常数$π（e）：=π（e，\ ell_ \ infty）$，定义是$ \ ell_ \ ell_ \ ell_ \ infty $的线性预测的最小规范。修复$ n \ geq 1 $，用$π_n$表示$π（\ cdot）$的最大值在$ n $二维真实的Banach空间中。我们证明，每一个$ \ varepsilon> 0 $都存在一个整数$ d \ geq 1 $和$ n $ - 二维的子空间$ e \ subset \ eeld \ ell_1^d $，以至于$π_n\ leqleqπ（e，\ ell_1^d）从$ \ lvert p \ rvert \leqπ（e，\ ell_1^d）+\ varepsilon $的意义上说，e $几乎是最小的。由于我们的主要结果，我们获得了一个公式，该公式将$π_n$与等级$ n $的正交投影矩阵的最小绝对值行和最小值。
The projection constant $Π(E):=Π(E, \ell_\infty)$ of a finite-dimensional Banach space $E\subset\ell_\infty$ is by definition the smallest norm of a linear projection of $\ell_\infty$ onto $E$. Fix $n\geq 1$ and denote by $Π_n$ the maximal value of $Π(\cdot)$ amongst $n$-dimensional real Banach spaces. We prove for every $\varepsilon >0$ that there exist an integer $d\geq 1$ and an $n$-dimensional subspace $E\subset\ell_1^d$ such that $Π_n \leq Π(E, \ell_1^d) +2 \varepsilon$ and the orthogonal projection $P\colon \ell_1^d\to E$ is almost minimal in the sense that $\lVert P \rVert \leq Π(E, \ell_1^d)+\varepsilon$. As a consequence of our main result, we obtain a formula relating $Π_n$ to smallest absolute value row-sums of orthogonal projection matrices of rank $n$.
