On Lattice Polyhedra and Pseudocircle Arrangements

Abstract

We study the maximum number of vertices fd(n) that a d-dimensional convex lattice polyhedron can have if the vertices have integer coordinates in the range {1, …, n}. We determine in dimension two for each k the smallest n such that {1, …, n}2 contains a convex 4k-gon, i.e. f2(n) = 4k, and construct upper and lower bounds for fd(n), d ≥ 3. The twodimensional case is used to construct a pseudocircle arrangement which shows that the bound of Clarkson e.a. on the number of incidences between points and pseudounitcircles is best possible. We construct for each n a strictly convex norm and a set of n points in the plane with cn4/3 unit distances with respect to that norm.