Recht-Ré noncommutative arithmetic-geometric mean conjecture is false

In this paper, we show that a conjecture of Recht and Ré is false. The conjecture is a noncommutative analogue of the arithmetic-geometric mean inequality where n positive numbers are replaced by $n$ positive defnite matrices. Our approach relies on the noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefnite program and the validity of the conjecture to certain bounds for the optimum values, which we show are false for $n = 5$. We also show $n = 2, 3$ cases are true.

Our method is nonconstructive. A paper by Christopher De Sa gives counterexamples for all $n \geq 5$. The $n = 4$ case remains unsolved.

The ICML version unfortunately contains a printing error in conjecture 1. Check the arXiv version for the correct statement.

Codes are available in https://github.com/laizehua/AM-GM-inequality.