Discrete Intrinsic Volumes and Grassmann Angle Valuations

For a convex lattice polytope P ⊂ \({\mathbb{R}}^{d}\) ${\mathbb{R}}^d$ of dimension d with vertices in \({\mathbb{Z}}^{d}\) ${\mathbb{Z}}^d$ , denote by L(P) its discrete volume which is defined as the number of integer points inside P. The classical Ehrhart theorem states that given a positive integer n, the function L(nP) is a polynomial in n of degree d, whose leading coefficient is the volume of P. In particular, L(nP) approximates the volume of nP for large n.

In convex geometry, one of the central notions generalizing the volume is the intrinsic volume. The main goal of the present paper is to introduce its discrete counterpart. In particular, it is proved that an analog of the Ehrhart theorem where volume is replaced by intrinsic volume holds true for it.

A notion of Grassmann angle valuation generalizing both discrete volume and the solid-angle valuation introduced by Reeve and Macdonald is also introduced and studied.