2021/12/15 by Pavle V. M. Blagojević, Blagojević, Pavle V. M., Paul Breiding +3
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2112.08437
openalex publication_date 2021/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers d≥ 2 and n≥ d+1 the configuration space of all facet volume vectors of all d-polytopes in \mathbb Rd with n facets is a full dimensional cone in \mathbb Rn. In particular, for tetrahedra (d=3 and n=4) this is a cone over a regular octahedron. Our proof is based on a novel configuration space / test map scheme which uses topological methods for finding solutions of a problem, and tools of differential geometry to identify solutions with the desired properties. Furthermore, our results open a possibility for the study of realization spaces of all d-polytopes in \mathbb Rd with n facets by the methods of algebraic topology.