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Consider the following set (spectrahedron/spectraheplex)

$$\mathcal A = \{ W : W \succeq 0, \mbox{tr}(W)=1 \}$$

Consider an approximating set

$$\mathcal B = \mbox{co} \{ u_i u_i^T : \|u_i\|_2 = 1, i=1,\dots,k \}$$

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.

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