Consider the following set (spectrahedron/spectraheplex)
$$\mathcal A = \left\{ W : W \succeq 0, \mbox{tr}(W)=1 \right\}$$
Consider an approximating set
$$\mathcal B = \mbox{co} \left\{ u_i u_i^T : \|u_i\|_2 = 1, i=1,\dots,k \right\}$$
the convex hull of a set of points where $u_i$ are uniformly sampled on the unit sphere.
I want to compute
$$\frac{\mathbb E[ \mbox{vol}(\mathcal B)] }{ \mbox{vol}(\mathcal A) }$$
as a function of $k$. Any ideas as to possible approaches, related literature, or related problems?
I’m completely stuck, though I feel it must be related to the expected volume of a randomly inscribed polygon in a circle, which I can compute.