Coefficients of the solid angle and Ehrhart quasi-polynomials

Abstract

Macdonald studied a discrete volume measure for a rational polytope $P$, called solid angle sum, that gives a natural discrete volume for $P$. We give a local formula for the codimension two quasi-coefficient of the solid angle sum of $P$. We also show how to recover the classical Ehrhart quasi-polynomial from the solid angle sum and in particular we find a similar local formula for the codimension one and codimension two quasi-coefficients. These local formulas are naturally valid for all positive real dilates of $P$. An interesting open question is to determine necessary and sufficient conditions on a polytope $P$ for which the discrete volume of $P$ given by the solid angle sum equals its continuous volume: $A_P(t) = \mathrm{vol}(P) t^d$. We prove that a sufficient condition is that $P$ tiles $\mathbb R^d$ by translations, together with the Hyperoctahedral group.