Pseudo-tetrahedral complexes

A pseudo-triangulation is a cell complex in the plane whose cells are pseudo-triangles, i.e., simple polygons with exactly three convex vertices (so-called corners). Being an interesting and flexible generalization of triangulations, pseudo-triangulations have found their place in computational geometry; see e.g. [8, 11, 7, 1] and references therein. Unlike triangulations, pseudo-triangulations eluded a meaningful generalization to higher dimensions so far. In this paper, we define pseudo-simplices and pseudo-simplicial complexes in d-space in a way consistent to pseudo-triangulations in the plane. Flip operations in pseudo-complexes are specified, as combinations of flips in pseudo-triangulations [11, 1], and of bistellar flips in simplicial complexes [9, 5, 4]. Our results are based on the concept of maximal locally convex functions on polyhedral domains [1], that allows us to unify several well-known structures, namely pseudo-triangulations, constrained Delaunay triangulations [3, 14], and regular simplicial complexes [2, 5]. Several implications of our results exist, and challenging open questions arise.