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

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.