  We introduce the set of framed convex polyhedra with N faces as the
symplectic quotient C^2N//SU(2). A framed polyhedron is then parametrized by N
spinors living in C^2 satisfying suitable closure constraints and defines a
usual convex polyhedron plus a phase for each face. We show that there is an
action of the unitary group U(N) on this phase space, which changes the shape
of faces and allows to map any polyhedron onto any other with the same total
area. This realizes the isomorphism of the space of framed polyhedra with the
Grassmannian space U(N)/SU(2)*U(N-2). We show how to write averages and
correlations of geometrical observables over the ensemble of polyhedra as
polynomial integrals over U(N) and we use the Itzykson-Zuber formula from
matrix models as the generating function for them. In the quantum case, a
canonical quantization of the framed polyhedron phase space leads to the
Hilbert space of SU(2) intertwiners. The individual face areas are quantized as
half-integers (spins) and the Hilbert spaces for fixed total area form
irreducible representations of U(N). We define coherent intertwiner states
peaked on classical framed polyhedra and transforming consistently under U(N)
transformations. The U(N) character formula can be considered as an extension
of the Itzykson-Zuber to the quantum level and generates the traces of all
polynomial observables over the Hilbert space of intertwiners. We finally show
that classical (convex) polygons can be described in a similar fashion trading
U(N) for O(N). We conclude with a discussion of the possible (deformation)
dynamics that one can define on the space of polygons or polyhedra. This work
is a priori useful in the context of discrete geometry but it should also be
relevant to (loop) quantum gravity in 2+1 and 3+1 dimensions when the quantum
geometry is defined in terms of gluing of (quantized) polygons and polyhedra.
