Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly distributed *on* a sphere;
(2) the convex hull of $N>n$ points randomly and uniformly distributed *in* a sphere;
(3) analogous definitions but using different distributions, or replacing "sphere" by "a given convex body."
I think my question is largely independent of the precise model:

Does the expected measure of the minimum face angle $\theta_{\min}$ over all faces of $P_n$ go to zero as $n \rightarrow \infty$?

I am hoping there is a succinct argument that avoids computing the precise expectation of $\theta_{\min}$, which might be difficult, and would certainly depend on the model. I have seen many papers on properties of random convex hulls, but none that I've found address my specific question. Thanks for ideas/pointers, under any model!