Strongly normal sets of contractible tiles in N dimensions

摘要

The second and third authors and others have studied collections of (usually) convex “tiles”—a generalization of pixels or voxels—in R2 and R3 that have a property called strong normality (SN): for any tile P, only finitely many tiles intersect P, and any nonempty intersection of those tiles also intersects P. This paper extends basic results about strong normality to collections of contractible polyhedra in Rn whose nonempty intersections are contractible. We also give sufficient (and trivially necessary) conditions on a locally finite collection of contractible polyhedra in R2 or R3 for their nonempty intersections to be contractible.