Determinant of random matrix, mixed volume of ellipsoids, and zeros of Gaussian random field

(joint work with Z. Kabluchko)
Consider $d\times d$ matrix $M$ whose columns are independent centered Gaussian vectors with covariance matrices $\Sigma_1,\dots,\Sigma_d$. Denote by $\mathcal{E}_i$ a location-dispersion ellipsoid of the $i$th column: $\mathcal{E}_i=\{t\in\mathbb{R}^d\,:\, t^\top\Sigma_i^{-1} t\leqslant1\}$. We show that
$$ \mathbb{E}\,|\det M|=c_{d}V(\mathcal{E}_1,\dots,\mathcal{E}_d), $$
where $V(\cdot,\dots,\cdot)$ denotes mixed volume. 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 $X=(X_1,\dots,X_k), k\leqslant d$. Using Kac-Rice formula, we obtain the geometric interpretation of the density of zeros of $X$ in terms of the mixed volume of location-dispersion ellipsoids of the gradients of $X_i/\sqrt{\mathrm{Var}(X_i)}$. This can be considered as a probabilistic analog of the well known Bernstein theorem about the number of the typical system of algebraic equations.
All necessary basic facts from geometry will be given.