Angle sums of simplicial polytopes

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.

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