Matrix multiplications and AM-GM inequalities

For positive-semidefinite d × d matrices A_1, …, A_n and any unitarily invariant norm |||·||| (including the operator norm and Schatten p-norms) and m ≤ n, define C(n,m,d) := inf frac 1/n^m Σ_j_1, j_2, …, j_m = 1^n |||A_j_1A_j_2… A_j_m||| (n-m)!/n! Σ_substackj_1, j_2, …, j_m = 1 \ all distinct^n…

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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@misc{cairn-matrix-multiplications-and-am-gm-inequalities,
  title        = {Matrix multiplications and AM-GM inequalities},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/matrix-multiplications-and-am-gm-inequalities}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

For positive-semidefinite $d \times d$ matrices $A_{1}, \dots, A_{n}$ and any unitarily invariant norm $∣∣∣ \cdot ∣∣∣$ (including the operator norm and Schatten $p$ -norms) and $m \leq n$ , define $C (n, m, d) : = in f \frac{\frac{1}{n ^{m}} \sum _{j_{1}, j_{2}, \dots, j_{m} = 1}^{n} ∣∣∣ A _{j_{1}} A _{j_{2}} \dots A _{j_{m}} ∣∣∣}{\frac{( n - m )!}{n !} \sum _{j_{1}, j_{2}, \dots, j_{m} = 1 all distinct}^{n} ∣∣∣ A _{j_{1}} A _{j_{2}} \dots A _{j_{m}} ∣∣∣}$ where the infimum is taken over all matrices $A_{1}, \dots, A_{n}$ and invariant norms $∣∣∣ \cdot ∣∣∣$ . What is $C (n, m, d)$ ?

What counts as progress

A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
A proof that a known construction is optimal, or a better bound on the other side.
Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.

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The problem

The problem

What counts as progress

Source and licence