  A toric polyhedron is a reduced closed subscheme of a toric variety that are
partial unions of the orbits of the torus action. We prove vanishing theorems
for toric polyhedra. We also give a proof of the $E_1$-degeneration of Hodge to
de Rham type spectral sequence for toric polyhedra in any characteristic.
Finally, we give a very powerful extension theorem for ample line bundles.
