Strongly normal sets of convex polygons or polyhedra

Abstract

A set P of nondegenerate convex polygons P in R 2 , or polyhedra P in R 3 , 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, P 1, …, P n , if each P i intersects P and I= P 1∩⋯∩ P n is nonempty, then I intersects P. The union of the P i∈ P that intersect P ∈ P is called the neighborhood of P in P , and is denoted by N P ( P). We prove that P is SN iff for any P ′ ⊆ P and P ∈ P ′, N P ′ ( 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 R 2 or R 3 into convex polygons or polyhedra are normal, but they may not be SN; for example, the square and hexagonal regular tessellations of R 2 are SN, but the triangular regular tessellation is not.