Angle sums of simplicial polytopes

TL;DR

This paper proves the non-negativity and certain symmetry properties of the $\uhat{eta}$-vector for simplicial polytopes, introduces a counterexample to unimodality, and extends results to cone valuations.

Contribution

It confirms the non-negativity and symmetry conjectures for the $\uhat{eta}$-vector and constructs a counterexample to unimodality, extending the theory of angle sums in polytopes.

Findings

Proved the non-negativity of the $\uhat{eta}$-vector.

Showed the $\uhat{eta}$-vector is increasing in the first half and flawless.

Constructed a 6D polytope with a non-unimodal $\uhat{eta}$-vector.

Abstract

The interior angle vector ($\widehat{\alpha}$-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 $\widehat{\alpha}$-vector were introduced by Sommerville (1927) and H\"ohn (1953). Camenga (2006) defined the $\widehat{\gamma}$-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 $\widehat{\gamma}$-vector increases in the first half and is flawless. In contrast to the $h$-vector, we construct a six-dimensional polytope with non-unimodal $\widehat{\gamma}$-vector. More generally, all result remain valid when solid angles are replaced by simple and non-negative cone valuations.