Some sharp inequalities for norms in $$\mathbb {R}^n$$ and $$\mathbb {C}^n$$

Stefan Gerdjikov, Nikolai Nikolov
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Abstract

The main result of this paper is that for any norm on a complex or real n-dimensional linear space, every extremal basis satisfies inverted triangle inequality with scaling factor \(2^n-1\). Furthermore, the constant \(2^n-1\) is tight. We also prove that the norms of any two extremal bases are comparable with a factor of \(2^n-1\), which, intuitively, means that any two extremal bases are quantitatively equivalent with the stated tolerance.

$$\mathbb {R}^n$$ 和 $$\mathbb {C}^n$$ 中规范的一些尖锐不等式
本文的主要结果是,对于复数或实数 n 维线性空间上的任意规范,每个极值基础都满足具有缩放因子 \(2^n-1\) 的倒三角不等式。此外,常数 \(2^n-1\) 是紧密的。我们还证明了任意两个极值基的规范是以\(2^n-1\)的因子进行比较的,直观地说,这意味着任意两个极值基在数量上是等价的,具有所述的容差。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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