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arXiv · 2606.23632

Sharp Inequalities for Products of Principal Minors of Positive Definite Matrices

Tobias Boege·Ludovick Bouthat
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Abstract

We study sharp inequalities for ratios of products of principal minors of real positive definite matrices. Our main result gives a closed-form solution to a family of nonconvex optimization problems over the positive definite cone. As a special case, we prove that the infimum of the Ingleton ratio over $4\times 4$ positive definite matrices is $16/27$, confirming a conjecture of Hall and Johnson. We also show that the cone of absolutely bounded ratios of products of principal minors is not polyhedral for $n\ge 4$, and that it is not semialgebraic over $\mathbb{Q}$.

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Tobias Boege, Ludovick Bouthat. 2026-06-22. Sharp Inequalities for Products of Principal Minors of Positive Definite Matrices. https://arxiv.org/abs/2606.23632

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