Infinite-dimensional conic Steiner formula

The classical Steiner formula expresses the volume of the neighborhood of a convex compact set in $\mathbb{R}^d$ as a polynomial in the radius of the neighborhood. In Tsirelson's work [16], this result was extended to the infinite-dimensional case. A spherical analogue of the Steiner formula for convex subsets of $\mathbb{S}^{d-1}$ is also well-known. The aim of this note is to obtain an infinite-dimensional version of this spherical analogue.