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What is Mixed Volume of Infinite-dimensional Convex Compact Sets about?

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.

Author

Dospolova, Mariia

Published

2023

Language

EN