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

## TL;DR

This paper explores special cases and dual perspectives of local formulas for Ehrhart coefficients, providing explicit descriptions and properties in symmetric and codimension one cases, advancing understanding of lattice tile methods.

## Contribution

It offers a dual viewpoint on the RS-μ construction for Ehrhart coefficients and analyzes its properties in specific symmetric and codimension one scenarios.

## Key findings

- Explicit dual description of the RS-μ construction.
- Properties of the construction in symmetric cases.
- Analysis of the construction in codimension one cases.

## 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.