Multidimensional Ehrhart Reciprocity

In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved that, if our polytope is a simplex, the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. In the present paper we complete the picture by extending this result to general rational polytopes. As a corollary, we also generalize a reciprocity theorem of Stanley.Comment: 7 page

Lattice points in vector-dilated polytopes

For $A\in\mathbb{Z}^{m\times n}$ we investigate the behaviour of the number of lattice points in $P_A(b)=\{x\in\mathbb{R}^n:Ax\leq b\}$, depending on the varying vector $b$. It is known that this number, restricted to a cone of constant combinatorial type of $P_A(b)$, is a quasi-polynomial function if b is an integral vector. We extend this result to rational vectors $b$ and show that the coefficients themselves are piecewise-defined polynomials. To this end, we use a theorem of McMullen on lattice points in Minkowski-sums of rational dilates of rational polytopes and take a closer look at the coefficients appearing there.Comment: 16 page

Integer hulls of linear polyhedra and scl in families

The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have eventually quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is eventually a ratio of quasipolynomials, and that unit balls in the scl norm eventually quasiconverge in finite-dimensional surgery families

A Closer Look at Lattice Points in Rational Simplices

We generalize Ehrhart's idea ([Eh]) of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n + 1 rational vertices, we use its description as the intersection of n+ 1 halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes ([Ma], [Mc], [St]). As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon