Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $\xi_1,\ldots,\xi_d$ with covariance matrices $\Sigma_1,\dots,\Sigma_d$. Denote by $\mathcal E_i$ the location-dispersion ellipsoid of $\xi_i$: $\mathcal E_i=\{\mathbf x\in\mathbb R^d\colon\mathbf x^\top\Sigma_i^{-1} \mathbf x\leqslant1\}$. We show that
$$ \mathbb E\,|\det M|=\frac{d!}{(2\pi)^{d/2}}V_d(\mathcal{E}_1,\dots,\mathcal E_d), $$
where $V_d(\cdot,\dots,\cdot)$ denotes the mixed volume. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an analytic expression for the mixed volume of $d$ arbitrary ellipsoids in $\mathbb R^d$.
As another application, we consider a smooth centered non-degenerate Gaussian random field $X=(X_1,\dots,X_k)^\top\colon\mathbb R^d\to\mathbb R^k$. Using the Kac–Rice formula, we obtain the geometric interpretation of the intensity of zeros of $X$ in terms of the mixed volume of location-dispersion ellipsoids of the gradients of $X_i/\sqrt{\mathbf{Var}X_i}$. This relates the zero sets of equations to the mixed volumes in a way which resembles the well-known Bernstein theorem on the number of solutions of a typical system of algebraic equations.