Sharp Inequalities for Products of Principal Minors of Positive Definite Matrices

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 $16\mathrm{/}27$, 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 $\mathbb{Q}$.