A solid angle theory for real polytopes

2007/07/31 by David DeSario, DeSario, David, Sinai Robins +1
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Computational Geometry and Mesh Generation #FOS: Mathematics #math.AC #math.CO

paper · pdf · doi:10.48550/arxiv.0708.0042

18 pages

arxiv created 2007/07/31 · openalex publication_date 2007/07/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Macdonald's solid-angle sum for real dilations of rational polygons
• Coefficients of the solid angle and Ehrhart quasi-polynomials
• Fourier transforms of polytopes, solid angle sums, and discrete volume
• Bounding minimal solid angles of polytopes
• Ehrhart-Macdonald reciprocity extended