论文标题
$ \ ell _ \ infty^n $的小型编成的大型等边集
Large equilateral sets in subspaces of $\ell_\infty^n$ of small codimension
论文作者
论文摘要
对于固定的$ k $，我们证明了$ \ ell _ {\ elfty}^n $的子空间的等边数的指数下限。特别是，我们表明，如果尺寸$ n $的规范空间的单位球是一个中央对称的多层，最多可与$ \ frac {4n} {3} {3} -o（n）$ facets对，那么它具有至少$ n+1 $的等边组合。其中包括$ \ ell _ {\ infty}^{n+2} $的$ 2 $ 2 $ 2 $ 2 $的子空间，用于$ n \ geq 9 $和codimension $ 3 $ of $ \ ell _ {\ infty}^{\ infty}^{n+3} $ for $ n \ geq 15 $。
For fixed $k$ we prove exponential lower bounds on the equilateral number of subspaces of $\ell_{\infty}^n$ of codimension $k$. In particular, we show that if the unit ball of a normed space of dimension $n$ is a centrally symmetric polytope with at most $\frac{4n}{3}-o(n)$ pairs of facets, then it has an equilateral set of cardinality at least $n+1$. These include subspaces of codimension $2$ of $\ell_{\infty}^{n+2}$ for $n\geq 9$ and of codimension $3$ of $\ell_{\infty}^{n+3}$ for $n\geq 15$.
