Angle sums of simplicial polytopes [article]

Sebastian Manecke
2020 arXiv   pre-print
The interior angle vector (α-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 α-vector were introduced by Sommerville (1927) and Höhn (1953). Camenga (2006) defined the γ-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
more » ... increases in the first half and is flawless. In contrast to the h-vector, we construct a six-dimensional polytope with non-unimodal γ-vector. More generally, all result remain valid when solid angles are replaced by simple and non-negative cone valuations.
arXiv:2007.07050v1 fatcat:mhsfmdlornabreww6pdmc3yws4