Strongly normal sets of convex polygons or polyhedra

A set V of nondegenerate convex polygons P in R, or polyhedra P in R, will be called normal if the intersection of any two of the P's of V is a face (in the case of polyhedra), an edge, a vertex, or empty. V is called strongly normal (SN) if it is normal and, for all P, Pi,..., Pn, if each P; intersects P and / = Px f) ... f~l Pn is nonempty, then I intersects P. The union of the Pt€ V that intersect P G V is called the neighborhood of P in V, and is denoted by NT(P). We prove that V is SN iff for any V C V and P € P', iV^(P) is simply connected. Thus SN characterizes sets V of polyhedra (or polygons) in which the neighborhood of any polyhedron, relative to any subset V of V, is simply connected. Tessellations of R or R into convex polygons or polyhedra are normal, but they may not be SN; for example, the square and hexagonal regular tessellations of R are SN, but the triangular regular tessellation is not. The support of the first author's research by the Army Medical Department under Grant DAMD17-971-7271, and of the second author's research by the Office of Naval Research under Grant N00014-95-1-0521, is gratefully acknowledged, as is the help of Janice Perrone in preparing this paper.