Steiner asked in 1832 what are the combinatorial types of convex polyhedra with their vertices on a quadric in 3-dimensional projective space. We will describe two recent advances on this problem.
One result (joint with Jeff Danciger and Sara Maloni) describes the combinatorial types of polyhedra inscribed in a one-sheeted hyperboloid or cylinder, while the other (joint with Hao Chen) deals with polyhedra having their vertices on a sphere in projective space which are not
contained in the ball.
The first result is based on anti-de Sitter geometry, while the second uses a natural extension of the hyperbolic space by the de Sitter space.