Discrete intrinsic volumes

Abstract

For a convex lattice polytope $P\subset \mathbb R^d$ of dimension $d$ with vertices in $\mathbb Z^d$, denote by $L(P)$ its discrete volume which is defined as the number of integer points inside $P$. The classical result due to Ehrhart says that for 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 notion which generalizes the volume is the intrinsic volumes. The main goal of this paper is to introduce their discrete counterparts. In particular, we show that for them the analogue of the Ehrhart result holds, where the volume is replaced by the intrinsic volume. We also introduce and study a notion of Grassmann valuation which generalizes both the discrete volume and the solid-angle valuation introduced by Reeve and Macdonald.