A conjecture on a tight norm inequality in the finite-dimensional l_p

TL;DR

The paper proposes a conjecture for a tight norm inequality in finite-dimensional l_p spaces, provides a proof for three dimensions, and confirms it numerically up to 200 dimensions, with relevance to quantum entropy minimization.

Contribution

It introduces a new conjecture for a tight norm inequality in l_p spaces, proves it for three dimensions, and verifies it numerically for higher dimensions.

Findings

Proof of the inequality for d=3

Numerical confirmation for d ≤ 200

Connection to quantum channel entropy minimization

Abstract

We suggest a tight inequality for norms in $d$-dimensional space $l_p $ which has simple formulation but appears hard to prove. We give a proof for $d=3$ and provide a detailed numerical check for $d\leq 200$ confirming the conjecture. We conclude with a brief survey of solutions for kin problems which anyhow concern minimization of the output entropy of certain quantum channel and rely upon the symmetry properties of the problem. Key words and phrases: $l_p $-norm, R\'enyi entropy, tight inequality, maximization of a convex function.