A Sharp Norm Inequality and Buzano's Inequality via Determinants

Abstract

We give a short linear-algebraic proof of the inequality $$\Vert x{\Vert }_1\Vert x{\Vert }_{\infty }\le \frac{1+\sqrt{n}}{2}\Vert x{\Vert }_2^2,$$ 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.