  We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which
are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces.
These metrics will be called ideal and hyper-ideal hyperbolic polyhedral
metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is
determined up to isometry by its curvature and a decorated ideal hyperbolic
polyhedral metric is determined up to isometry and change of decorations by its
curvature. The main tool used in the proof is the Fenchel dual of the volume
function.

Geometric Topology