On the necessary and sufficient conditions for Hadamard-Fischer-Koteljanskii type inequalities

This work explores the ratios of products of determinants of principal submatrices of positive definite matrices. We investigate conditions under which these ratios are bounded, particularly revisiting the necessary/sufficient conditions proposed by Johnson and Barrett. This analysis extends to set-theoretic consequences and unboundedness of certain ratios. We also demonstrate how these conditions can be used to prove the boundedness of several known determinantal inequalities. Additionally, we address the optimization problem of finding the supremum of such ratios over all positive definite matrices, formulating it as a linear optimization program. Finally, for completeness, we include the proofs of theorems that appear to have been previously known but lack accessible proofs.