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 $\{({\xi }_1(h),\dots ,{\xi }_k(h)):h\in K\}\subset {\mathbb{R}}^k,$ where ${\xi }_1,\dots ,{\xi }_k$ 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{(2π)^{k/2}}{k!κ_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.

23 pages. Published: M. K. Dospolova, Mixed volume of infinite-dimensional convex compact sets, Probability and statistics. Part 32, Zap. Nauchn. Sem. POMI, 510, POMI, St. Petersburg, 2022, 98-123