小余维$\ell_\infty^n$子空间中的大等边集
Large equilateral sets in subspaces of $\ell_\infty^n$ of small codimension
摘要：
对于固定的$k$，我们证明了余维$k$的$\ell_{\infty}^n$的等边子空间个数的指数下界。特别地，我们证明了，如果单位球的赋范空间的尺寸$n$是一个中心对称多面体与最多$\frac{4n}{3}-o(n)$对的方面，那么它有一个等边集的基数至少$n+1$。这些包括用于$n\geq 9$的$\ell_{\infty}^{n+2}$的余维$2$的子空间和用于$n\geq 15$的$\ell_{\infty}^{n+3}$的余维$3$的子空间。
英文摘要：
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 $n\geq 15$.
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