AlphaEvolve problems·alphaevolve:47

Matrix multiplications and AM-GM inequalities

For positive-semidefinite $d \\times d$ matrices $A_1, \\ldots, A_n$ and any unitarily invariant norm $|||\\cdot|||$ (including the operator norm and Schatten $p$-norms) and $m \\leq n$, define $$ C(n,m,d) \\coloneqq \\inf \\frac{ \\frac{1}{n^m} \\sum_{j_1, j_2, \\ldots, j_m = 1}^{n} |||A_{j_1}A_{j_2}\\ldots A_{j_m}|||}{ \\frac{(n-m)!}{n!} \\sum_{\\substack{j_1, j_2, \\ldots, j_m = 1 \\\\ \\text{all distinct}}}^{n} |||A_{j_1}A_{j_2}\\ldots A_{j_m}|||} $$ where the infimum is taken over all matrices $A_1,\\dots,A_n$ and invariant norms $|||\\cdot|||$. What is $C(n,m,d)$?

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