It is possible to see any eleven
vertices of an opaque solld 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 vlsibility properties in Euclidean space E2 and Er. 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 E3 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
E2 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 (V1 and V2) 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 4n vertices can see externally simultaneously. It
will be shown that this polyhedron can be constructed for arbitrarily large
n.
