2020/07/14 by Sebastian Manecke, Manecke, Sebastian
Mathematics · #05E45 #52B05 #52B11 #52B45 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2007.07050
openalex publication_date 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The interior angle vector (\widehatα-vector) of a polytope is a metric analogue of the f-vector in which faces are weighted by their solid angle. For simplicial polytopes, Dehn-Sommerville-type relations on the \widehatα-vector were introduced by Sommerville (1927) and Höhn (1953). Camenga (2006) defined the \widehatγ-vector, a linear transformation analogous to the h-vector and conjectured it to be non-negative. Using tools from geometric and algebraic combinatorics, we prove this conjecture and show that the \widehatγ-vector increases in the first half and is flawless. In contrast to the h-vector, we construct a six-dimensional polytope with non-unimodal \widehatγ-vector. More generally, all result remain valid when solid angles are replaced by simple and non-negative cone valuations.