More on Finite Subsets and Simple Closed Polygonal Paths

Abstract

Since F is finite we can find parallel hyperplanes 7r and 7r' such that F is contained between 7r and 7r'. (Figure 1 gives the picture in going from R2 to R3.) Let T be the set of points determined by the intersection with 7r' of all lines in Rmn which contain two or more points of F. Let z be any point in 7r'T and K(F) be the set of all segments joining pairs of points in F. The region bounded by 7r and 7r' is convex since it is the intersection of two convex regions. Therefore K(F) also lies in this region. Let Jz be the projection of K(F) onto 7r from z, i.e., if yEK(F), then jz(y) is the intersection with 7r of the line determined by y and z. Since the points of F are not all coplanar, the points of j2(F) are not all collinear, for if they were then F would be in the plane determined by j,(F) and z. Since z C7r'-T, j4 F is 1-1, and the image of K(F) is the set of all segments joining pairs of points in jz(F). By the induction hypothesis, j2(K(F)) contains a s.c.p.p. C for which j,(F) is the set of vertices. Then j7-(C) is a s.c.p.p. for which F is the set of vertices. If x and y are distinct points of the m-sphere Sm, a segment joining x and y