Mixed volume of infinite-dimensional convex compact sets

Abstract

Let $K$ be a convex compact $GB$-subset of a separable Hilbert space $H$. Denote by $\mathrm{Spec}_k K$ the set $\{(\xi_1(h), \ldots, \xi_k(h))\colon h\in K\}\subset \mathbb{R}^k,$ where $\xi_1, \ldots, \xi_k$ are independent copies of the isonormal Gaussian process on $H$. Tsirelson showed that in this case the intrinsic volumes of $K$ satisfy the relation \begin{equation*} V_k(K)= \frac{(2\pi)^{k/2}}{k!\kappa_k} \mathbf{E}\,\mathrm{Vol}_k(\mathrm{Spec}_k K). \end{equation*} Here, $\mathbf{E} \ \mathrm{Vol}_k(\mathrm{Spec}_k K)$ is the mean volume of $\mathrm{Spec}_k K$ and $\kappa_k$ is the volume of the $k$-dimensional unit ball. In this work, we generalize Tsirelson's theorem to the mixed volumes of the infinite-dimensional convex compact $GB$-subsets of $H$, first introducing this notion. Moreover, using the obtained result we compute the mixed volume of the closed convex hulls of the two orthogonal Wiener spirals.