Duchi’s noncommutative AM–GM conjecture (matrix products)

Prove that for positive-semidefinite d×d matrices A_1,…,A_n, all m ≤ n, and any unitarily invariant norm, the averaged product inequality achieves equality C_amgm(n,m,d) = 1; in particular, settle the case m ≥ 4.

Background

The conjecture generalizes AM–GM-type inequalities to noncommutative matrix products under unitarily invariant norms. Cases m = 1,2 and m = 3 have been proved; higher m remains open.

A resolution would bridge convexity inequalities, operator theory, and algorithmic implications for matrix computations.

References

Duchi conjectured that $ C_{\ref{amgm}(n,m,d)=1$ for all $n,m,d$. The cases $m=1,2$ of this conjecture follow from standard arguments, whereas the case $m=3$ was proved in . The case $m \geq 4$ is open.

— Mathematical exploration and discovery at scale (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Matrix multiplications and AM–GM inequalities” (Section 4.31)