Eventual polynomiality and dualities of relative Ehrhart functions

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 $\mathrm{e}\mathrm{h}\mathrm{r}(P,Q;t)$ of such valid translations of $Q$ into $tP$ inherits eventual quasi-polynomiality (or eventual polynomiality) with leading term $\mathrm{v}\mathrm{o}\mathrm{l}(P)t^d$, 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. Similarly, replacing $P$ with its relative interior $P^{\circ }$ defines ${\mathrm{e}\mathrm{h}\mathrm{r}}^{\circ }(P,Q;t)$, which is also shown to be an eventual quasi-polynomial (or eventual polynomial). Furthermore, we prove several formulas for relative Ehrhart functions under the condition that $P=k{P}_0$ and $Q=l{Q}_0$ for some integers $k$, $l>0$, and polytopes $P_0$ and $Q_0$ with $\dim Q_0>0$ such that $Q_0$ is inscribed in $P_0$. In particular, we prove that under this condition, $\mathrm{e}\mathrm{h}\mathrm{r}(P,Q;-t)=(-1{)}^d{\mathrm{e}\mathrm{h}\mathrm{r}}^{\circ }(P,Q;t+\rho )$ $(t\gg 0)$ holds for some integer $\rho >0$ if and only if $k\mid 2l$ via the classical Ehrhart–Macdonald reciprocity law. We also prove that under the same condition, $\mathrm{e}\mathrm{h}\mathrm{r}(P,Q;t)={\mathrm{e}\mathrm{h}\mathrm{r}}^{\circ }(P,Q;t+\sigma )$ $(t\gg 0)$ holds for some integer $\sigma >0$ if and only if ${\mathrm{e}\mathrm{h}\mathrm{r}}^{\circ }(P_0;\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{g}{P}_0)=1$ and $k\mid \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{g}{P}_0$.