Special cases and a dual view on the local formulas for Ehrhart coefficients from lattice tiles

Abstract

McMullen's formulas or local formulas for Ehrhart coefficients are functions on rational cones that determine the $i$-th coefficient of the Ehrhart polynomial as a weighted sum of the volumes of the i-dimensional faces of a polytope. This work focuses on the RS-$\mu$-construction as given in a previous paper by Achill Sch\"urmann and the author. We give an explicit description of the construction from the dual point of view, i.e. given the cone of feasible directions instead of the normal cone as input value. We further show some properties of the construction in special cases, namely in case of symmetry and for the codimension one case.