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arXiv · 2303.03210

Some 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.

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Stefan Gerdjikov, Nikolai Nikolov. 2023-03-06. Some inequalities for norms in $\mathbb{ R}^n$ and $\mathbb{ C}^n$. https://doi.org/10.1007/s00605-024-02004-7

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