Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

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 ?(nm ^{n-1 1} ) and O(n²m ^{n-1 1} + n⁵4ⁿ) with high probability when m = 2^?(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 1 T(n²m ^n-1 ) 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 ?(m ^n-1).