  We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices
on a sphere. We do not require, in particular, that the polyhedra lie in the
interior of the sphere, hence the term "weakly inscribed". Such polyhedra can
be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a
combination of the hyperbolic space and the de Sitter space, with the sphere as
the common ideal boundary. We have three main results: (1) the $1$-skeleta of
weakly inscribed polyhedra are characterized in a purely combinatorial way, (2)
the exterior dihedral angles are characterized by linear programming, and (3)
we also describe the hyperbolic-de Sitter structure induced on the boundary of
weakly inscribed polyhedra.

Metric Geometry;Differential Geometry;Geometric Topology