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.

arXiv comment39 pages, 5 figures

References in corpus (7)

• Two Dimensional Gauge Theories Revisited
• Multi-tiling and equidecomposability of polytopes by lattice translates
• Fourier transforms of polytopes, solid angle sums, and discrete volume
• Reconstruction of rational polytopes from the real-parameter Ehrhart function of its translates
• Semi-reflexive polytopes
• Convergence of multiple Fourier series and Pick's theorem
• Macdonald's solid-angle sum for real dilations of rational polygons

Cited by in corpus (1)

• A friendly introduction to Fourier analysis on polytopes