Mixed Volume of Infinite-Dimensional Convex Compact Sets

Let K be a convex compact GB-subset of a separable Hilbert space H. Denote by Spec_kK the set {(ξ₁(h), . . . , ξ_k(h)) : h ∈ K} ⊂ \({\mathbb{R}}^{k},\) ${\mathbb{R}}^k,$ where ξ₁, . . . , ξ_k are independent copies of the isonormal Gaussian process on H. Tsirelson showed that the intrinsic volumes of K in this case satisfy the relation

\({V}_{k}\left(K\right)=\frac{{\left(2\pi \right)}^{k/2}}{k!{\kappa }_{k}}{\mathrm{E\;Vol}}_{k}\left({\mathrm{Spec}}_{k}K\right).\) $V_k\left(K\right)=\frac{{\left(2,\pi \right)}^{k/2}}{k!{\kappa }_k}{\mathrm{E}\mathrm{Vol}}_k\left({\mathrm{Spec}}_k,K\right).$

Here E Vol_k(Spec_kK) is the mean volume of Spec_kK and κ_k is the volume of the k-dimensional unit ball.

In this paper, 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 result obtained we compute the mixed volume of the closed convex hulls of the two orthogonal Wiener spirals.