  There are (at least) two reasons to study random polytopes. The first is to
understand the combinatorics and geometry of random polytopes especially as
compared to other classes of polytopes, and the second is to analyze
average-case complexity for algorithms which take polytopal data as input.
However, establishing results in either of these directions often requires
quite technical methods. Here we seek to give an elementary introduction to
random polytopes avoiding these technicalities. In particular we explore the
general paradigm that polytopes obtained from the convex hull of random points
on a sphere have low complexity.
