Stresses and Liftings of Cell-Complexes

Abstract

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

On Traces ofd-stresses in the Skeletons of Lower Dimensions of Piecewise-lineard-manifolds

“…In particular, a segment whose ends are the vertices of the reciprocal corresponding to two adjacent d-cells of is perpendicular to their common facet. This oneto-one correspondence holds only when certain homological restrictions are placed on the manifold, for example, when H 1 ( , Z 2 ) = 0 [19]. Notice that the star of a cell satisfies this condition.…”

“…This generalization is useful in the combinatorics and geometry of piecewise-linear manifolds, rigidity theory, and the theory of Dirichlet-Voronoi diagrams. Such generalizations have been proposed by Lee [13], Tay et al [24], Crapo and Whiteley [8], and Rybnikov [19,20].…”

“…Their idea was, perhaps, motivated by Minkowski's theorem on the vanishing of the sum of normals to a convex polytope at its facets (Theorem 5.1 of Section 5). The projections of dmanifolds with trivial H 1 over Z 2 from R d+1 onto R d correspond to d-stresses (see Section 2 and [19,29]); thus, one can reformulate their conjecture as the existence of a natural correspondence between d-stresses on a d-manifold realized in R d and stresses on its 1-skeleton (in the general theory of stresses such stresses are called 2-stresses). Theorem 6.2 estublishes a natural one-way connection between d-stresses and k-stresses (k ≤ d) for oriented piecewiselinear manifolds realized in R d .…”

“…In this case the equilibrium of forces is required not at each vertex, but at each (d − 2)-cell [6,19,26]. Such stresses are called d-stresses because d is the lowest dimension of a manifold for which the space of d-stress is non-trivial in the sense that it essentially depends on the combinatorics and geometry of the manifold.…”

A Logarithmic-Time Solution to the Point Location Problem for Closed-Form Linear MPC

“…Although it is not difficult to find problems for which a lifting does not exist, general conditions for the existence of a lifting for quadratic costs are not known. See (Aurenhammer, 1991;Rybnikov, 1999) for details on testing when a complex has an appropriate lifting.…”