A Sharp Norm Inequality and Buzano's Inequality via Determinants

Abstract:We give a short linear-algebraic proof of the inequality $$ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{n}}{2}\,\|x\|_2^2, $$ valid for every $x\in\mathbb{R}^n$. 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 $(1+\sqrt{n})/2$ is optimal. The inequality is a special case of Buzano's inequality, and the same method also proves Buzano's inequality for real vectors.

Classical Analysis and ODEs (math.CA); Machine Learning (cs.LG); History and Overview (math.HO); Optimization and Control (math.OC)

arXiv:2604.01525 [math.CA]

(or arXiv:2604.01525v2 [math.CA] for this version)

https://doi.org/10.48550/arXiv.2604.01525 Focus to learn more arXiv-issued DOI via DataCite

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