Relative Ehrhart theory I: relative Ehrhart eventual polynomials

Abstract

Classical Ehrhart theory measures the discrete capacity of a convex rational (or integral) polytope $P$ by counting the number of lattice points in the $t$-th dilate $tP$ of $P$. In this paper, we extend this paradigm by replacing a lattice point with a geometric object $Q$ of dimension at most $\dim P$. We show that the counting function $\mathrm{ehr}(P,Q;t)$ of such valid translations of $Q$ into $tP$ inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term $\mathrm{vol}(P)t^d$, where $d=\dim P$. This result is naturally derived by induction on the dimension, based on the classical quasi-polynomiality (or polynomiality) of Ehrhart functions.

10 pages, 6 figures; Minor corrections and refined notation