极大相对投影常数的稳定性
Stability of maximal relative projection constants
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中文总结 AI 辅助
该论文针对$\ell_\infty^n$子空间的极大相对投影常数稳定性问题，证明当$n\ge 2^r\binom{r+1}{2}$时$\lambda(r,n)$稳定于$\lambda(r)$，解答了Basso的问题。
中文摘要 AI 辅助
对于正整数$n\ge r$，令$\lambda(r,n)$表示$\ell_\infty^n$中$r$维子空间的极大相对投影常数，$\lambda(r)$表示极大绝对投影常数。已知对任意固定的$r$，$\lambda(r,n)$是单调非减序列，当$n\to\infty$时极限为$\lambda(r)$。自然的问题是：是否存在$n>r$使得$\lambda(r,n)$稳定于$\lambda(r)$？我们证明，对任意固定的$r$，当$n\ge 2^r\binom{r+1}{2}$时，$\lambda(r,n)=\lambda(r)$，这回答了Basso提出的问题，所用技术具有独立意义。
英文摘要
For positive integers $n\ge r$, let $λ(r,n)$ denote the \emph{maximal relative projection constant} of $r$-dimensional subspaces of $\ell_\infty^n$ and $λ(r)$ denote the \emph{maximal absolute projection constant}, respectively. It is known that for any fixed $r$, $λ(r,n)$ is a non-decreasing sequence with limit $λ(r)$ as $n\to \infty$. A natural question is whether $λ(r,n)$ stabilizes at $λ(r)$ for some $n>r$. We prove that for any fixed $r$, \[λ(r,n)=λ(r) \qquad \text{for every}\qquad n\ge 2^{r}\binom{r+1}{2}.\] This answers a question of Basso. The technique used is of independent interest.
