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 Ω(nm 1 n−1 ) and O(nm 1 n−1 + n4) with high probability when m ≥ 2. 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(nm 1 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 Ω(m 1 n−1 ). ∗g.f.y.bonnet@rug.nl; Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, The Netherlands; CogniGron (Groningen Cognitive Systems and Materials Center), University of Groningen, The Netherlands. †Partially funded by the CogniGron research center and the Ubbo Emmius Funds (Univ. of Groningen). ‡Partially funded by the DFG Priority Program (SPP) 2265 Random Geometric Systems, project P23. §dadush@cwi.nl; Centrum Wiskunde & Informatica, The Netherlands ¶Supported by the ERC Starting grant QIP–805241. ‖uri.grupel@uibk.ac.at; University of Innsbruck, Austria ∗∗s.huiberts@cwi.nl; Centrum Wiskunde & Informatica, The Netherlands ††glivshyts6@math.gatech.edu; Georgia Institute of Technology, United States 1 ar X iv :2 11 2. 13 02 7v 1 [ m at h. PR ] 2 4 D ec 2 02 1