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Asymptotic bounds on the combinatorial diameter of random polytopes
The combinatorial diameter of a polytope 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 -dimensional polytope defined by the intersection of i.i.d.\
half-spaces whose normals are chosen uniformly from the sphere, we show that
is and with high probability when .
For the upper bound, we first prove that the number of vertices in any fixed
two dimensional projection sharply concentrates around its expectation when
is large, where we rely on the 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 , corresponding to a random convex hull, by showing the
relation .
We then prove that the shortest path between any ``nearly'' antipodal pair
vertices of has length