On the polyhedral complexity of the integer points in a hyperball

Let Bn be a hyperball in Rn, n≥2, and denote . Define polyhedral facet complexity of as where P is an enclosing polyhedron for (i.e., ) and fn−1(P) is the number of the (n−1)-facets of P. Analogously, define polyhedral vertex complexity of as where P is an enclosing polyhedron for and f0(P) is the number of the 0-facets (vertices) of P. Upper bounds for follow from a well-known bound for the number of facets and vertices of the convex hull of [I. Bárány, D.G. Larman, The convex hull of the integer points in a large ball, Math. Ann. 312 (1998) 167–181]. In this note we provide the first nontrivial lower bounds on and .