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

TL;DR

This paper presents a concise linear algebra proof of a sharp inequality relating , , and  norms in finite-dimensional spaces, with applications in optimization and numerical analysis.

Contribution

It introduces a novel determinantal approach to prove the inequality and establishes the optimality of the constant involved.

Findings

The inequality   norms holds with constant +\u221a/2 for all vectors.

The proof exploits the determinantal structure of quadratic forms.

The constant +/2 is proven to be optimal.

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.