Titile: Scribability problems for polytopes
Abstract: The story begins in 1832 when Steiner asked about the existence of non-inscribable 3-dimenional polytopes. The first example was not given until nearly 100 years later by Steinitz. Then the scribability problem is generalized to polytopes in higher dimensions with respect to faces other than vertices and facets, and several weakened versions are proposed. We propose a new variation, namely the (i,j)-scribability problem, in strong and weak forms. Roughly speaking, we wonder: can every d-dimensional polytope be realized with all their i-faces "outside" the sphere, while all their j-faces "intersect" the sphere? We managed to give answers to most cases. The (i,j)-scribability is interesting in its own right, and also helpful for classical scribability problems. In particular, we obtain complete answer to the classical scribability problem for stacked polytopes and cyclic polytopes. This is a joint work with A. Padrol.