The convex hull of the integer points in a large ball

@inproceedings{Brny1998TheCH,
title={The convex hull of the integer points in a large ball},
author={Imre B{\'a}r{\'a}ny and David G. Larman},
year={1998}
}

wherefk(P) denotes the number of k–dimensional faces of the polytope P. The limit, as R → ∞, of the average of r −2/3f0(Pr ), on an interval [ R,R + H ], is determined by Balog and Deshoullier [BD], and turns out to be 3 .45 . . . , (H must be large). Our main result extends (1.1) to any d ≥ 2 and to anyfk(Pr ) with k = 0, . . . ,d − 1. Theorem 1. For every d≥ 2 there are constants c 1(d) and c2(d) such that for all k ∈ {0, . . . ,d − 1}