External visibility
Edwin O. Buchman, F. A. Valentine · Pacific Journal of Mathematics · 1976
It is possible to see any eleven vertices of an opaque solid regular icosahedron from some appropriate point, although it is not possible to see all twelve vertices simultaneously.In this paper we refer to visibility in the complement of a convex set as external visibility.Valentine has investigated external visibility properties in Euclidean space E 2 and E\ One question raised was the following: does there exist a fixed number h such that if every h vertices of an arbitrary bounded closed convex polyhedron in E 3 can see some common point externally, then all the vertices can see some common point externally?The answer is no, which is surprising since the corresponding question in E 2 will be answered affirmatively, with h = 5. Figure 2 illustrates a solid convex polyhedron with 4n vertices (n = 4 in the illustration), where the top and bottom vertices (V ] and V 2 ) have valence 2n -1.Each collection of 4n -1 vertices of this polyhedron can see some point in space externally, yet there is no point which all An vertices can see externally simultaneously.It will be shown that this polyhedron can be constructed for arbitrarily large n.