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

2026-04-01

AI Summary

This study addresses sharp inequalities among the $\ell^1$, $\ell^2$, and $\ell^\infty$ norms in finite-dimensional spaces. By constructing a parameterized quadratic form and analyzing its determinant structure, the authors introduce a purely linear-algebraic approach that reveals the intrinsic algebraic mechanism underlying these norm relations. This method yields a concise derivation of the optimal inequality $\|x\|_1 \leq \frac{1+\sqrt{p}}{2} \|x\|_2 + \frac{1+\sqrt{p}}{2} \|x\|_\infty$ for all $x \in \mathbb{R}^p$, along with its equivalent formulations. Moreover, the constant $(1+\sqrt{p})/2$ is rigorously shown to be optimal, attaining the theoretical lower bound.

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.

Problem

• norm inequality
• ell^1 norm
• ell^infty norm
• ell^2 norm
• optimal constant

Innovation

• determinantal structure
• norm inequality
• quadratic forms
• optimal constant
• linear algebra