A solid angle theory for real polytopes

TL;DR

This paper generalizes solid angle sum theorems from rational to real polytopes and extends Macdonald's quasipolynomial to a real analytic function of the dilation parameter.

Contribution

It introduces a framework for analyzing solid angle sums over real polytopes and real dilation parameters, extending classical results to a broader setting.

Findings

Extended solid angle sum theorems to real polytopes

Developed a real analytic extension of Macdonald's quasipolynomial

Applicable to any real convex polytope

Abstract

We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation parameters. One of the main results is an extension of Macdonald's solid angle quasipolynomial for rational polytopes to a real analytic function of the dilation parameter, for any real convex polytope.