Stresses and liftings of cell-complexes

This paper introduces a general notion of stress on cell-complexes and repo rts on connections between stresses and liftings (generalization of C-1(0)- splines) of d-dimensional cell-complexes in R-d. New sufficient conditions for the existence of a sharp lifting for a "flat" piecewise-linear realizat ion of a manifold are given. Our approach also gives some new results on th e equivalence between spherical complexes and convex and star polytopes. As an application, two algorithms ate given that determine whether a piecewis e-linear realization of a d-manifold in R-d admits a lifting to Rd+1 which satisfies given constraints. We also demonstrate connections between stress es and Voronoi-Dirichlet diagrams and show that any weighted Voronoi-Dirich let diagram without non-compact cells can be represented as a weighted Dela unay decomposition and vice versa.