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arXiv · 2604.01525

A Determinantal Approach to a Sharp $\ell^1-\ell^\infty-\ell^2$ Norm Inequality

Jose Antonio Lara Benitez
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Abstract

We give a short linear--algebraic proof of the inequality \[ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{p}}{2}\,\|x\|_2^2, \] valid for every \(x\in\mathbb{R}^p\). This inequality relates three fundamental norms on finite-dimensional spaces and has applications in optimization and numerical analysis. Our proof exploits the determinantal structure of a parametrized family of quadratic forms, and we show the constant $(1+\sqrt{p})/2$ is optimal.

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BibTeXRIS

Jose Antonio Lara Benitez. 2026-04-02. A Determinantal Approach to a Sharp $\ell^1-\ell^\infty-\ell^2$ Norm Inequality. https://arxiv.org/abs/2604.01525

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