  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.
