The combinatorial diameter diam(P) of a polytope P is the
maximum shortest path distance between any pair of vertices. In this paper, we
provide upper and lower bounds on the combinatorial diameter of a random
"spherical" polytope, which is tight to within one factor of dimension when the
number of inequalities is large compared to the dimension. More precisely, for
an n-dimensional polytope P defined by the intersection of m i.i.d.\
half-spaces whose normals are chosen uniformly from the sphere, we show that
diam(P) is Ω(nmn−11) and O(n2mn−11+n54n) with high probability when m≥2Ω(n).
For the upper bound, we first prove that the number of vertices in any fixed
two dimensional projection sharply concentrates around its expectation when m
is large, where we rely on the Θ(n2mn−11) bound on the
expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter
upper bound, we stitch these ``shadows paths'' together over a suitable net
using worst-case diameter bounds to connect vertices to the nearest shadow. For
the lower bound, we first reduce to lower bounding the diameter of the dual
polytope P∘, corresponding to a random convex hull, by showing the
relation diam(P)≥(n−1)(diam(P∘)−2).
We then prove that the shortest path between any ``nearly'' antipodal pair
vertices of P∘ has length Ω(mn−11)