Strongly normal sets of convex polygons or polyhedra

Abstract

A set P of nondegenerate convex polygons P in R2, or polyhedra P in R3, will be called normal if the intersection of any two of the Ps of P is a face (in the case of polyhedra), an edge, a vertex or empty. P is called strongly normal (SN) if it is normal and, for all P, P1, ..., Pn, if each Pi intersects P and I=P1∩∩Pn is nonempty, then I intersects P. The union of the Pi∈P that intersect P ∈ P is called the neighborhood of P in P, and is denoted by NP(P). We prove that P is SN iff for any P′ ⊆ P and P ∈ P′, NP′(P) is simply connected. Thus SN characterizes sets P of polyhedra (or polygons) in which the neighborhood of any polyhedron, relative to any subset P′ of P, is simply connected. Tessellations of R2 or R3 into convex polygons or polyhedra are normal, but they may not be SN; for example, the square and hexagonal regular tessellations of R2 are SN, but the triangular regular tessellation is not.