I am thinking about the following problem:
Let $$M = \begin{bmatrix}A & X_{12} & X_{13} \\ X_{21} & B & X_{23} \\ X_{31} & X_{32} & C\end{bmatrix} \in \mathbb{R}_{\geq0}^{n \times n},$$ and define the block (sub-)matrices $$M_1 = \begin{bmatrix}A & X_{12} \\ X_{21} & B\end{bmatrix},\quad M_2 = \begin{bmatrix}A & X_{13} \\ X_{31} & C\end{bmatrix},\quad M_3 = \begin{bmatrix}B & X_{23} \\ X_{32} & C\end{bmatrix}.$$ Suppose the matrices $M, M_1, M_2, M_3, A, B, C$ are symmetric and positive definite.
Does the following hold? $$\frac{|M_1| \cdot |M_2| \cdot |M_3|}{|M|} \leq |A| \cdot |B| \cdot |C|$$ where $|\cdot|$ denotes the determinant. One can also write this in terms of the Schur complement: $$|M_1 / B| \cdot |M_2 / C| \leq |A| \cdot |M / M_3|.$$
I tried finding a counterexample, but nothing came to mind. Do you have an idea? :)
$\begin{bmatrix}1 & 0.4 & 0.001 \\ 0.4 & 1 & 0.4 \\ 0.001 & 0.4 & 1\end{bmatrix}$