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

Jose Antonio Lara Benitez

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.

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

Abstract

We give a short linear--algebraic proof of the inequality valid for every . 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 is optimal.

Paper Structure

This paper contains 3 sections, 19 equations.

Table of Contents

  1. Introduction
  2. Proof of the Inequality
  3. Optimality of the Constant

Theorems & Definitions (4)

  • Claim 1
  • proof
  • Claim 2
  • proof