  Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which
can be either uniformly sampled or a Poisson point process. Its convex hull is
a random inscribed polytope, whose boundary approximates the sphere. We focus
on the case $d=3$, for which there are elementary proofs and fascinating
formulas for metric properties. In particular, we study the fraction of acute
facets, the expected intrinsic volumes, the total edge length, and the distance
to a fixed point. Finally we generalize the results to the ellipsoid with
homeoid density.

Metric Geometry;Probability