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A Sharp Norm Inequality and Buzano's Inequality via Determinants

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…

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2026
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arxiv.org/abs/2604.01525
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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.