Old Topology Problem

'Gazeta Matematica', the most widely known Mathematical journal in Romania, begins its year by publishing problems from old years ending in the same digit as the year that just started. This year, they published problems from years ending in 8. The problem that I am about to share was published in 1968 by T. Zamfirescu, and is solvable through Helly's theorem. Without further ado, let's get straight into the problem.

Let d be a line in space and S a finite set of spheres such that any 3 spheres have an interior point in common. Prove that there exists a line d' parallel to d that intersects all the spheres.

Helly's theorem in a plane states that if a finite number of convex sets with the property that any three convex sets intersect, then all convex sets intersect.

Here's the proof of the theorem: Let m be smaller or equal to n, where m is the maximum number of sets with the intersection I different than the empty set and let us presume by absurd that \(m<n\), so there is C such that C doesn't intersect with I. Because C and I are convex sets, there must be a line d that separates them, they are in planes \(S_1\) and \(S_2\) respectively. Because \(m>2\), we need at least 3 sets to create I. Let \(C_1\) and \(C_2\) be two of them. Obviously, I is included in them. From the hypothesis, \(C_1\), \(C_2\) and C intersect. We choose the point A in their intersection and the point B in I. Thus, A is in \(S_1\) and B is in \(S_2\). Since A and B are in \(C_1\) and \(C_2\), the whole segment AB is in \(C_1\) and \(C_2\), and, thus, AB and d intersect. We now have that for any sets \(C_1\) and \(C_2\) that contain I, the intersection between \(C_1, C_2\) and d is not the empty set. By writing in a different form, we have that the intersection between \(C_1\) and d intersects with the intersection between $C_2$ and d. Thus, by considering the intersections as segments, we have m segments on a line such that each pair of segments intersects. By Helly's theorem on a line, we have that all the m segments intersect, so, by writing the intersection of the segments as the intersection of \(C_i\)-s and d, we have that I intersects with d, which is a contradiction. Thus, m=n.

Back to the problem at hand, let us consider a plane P such that d is perpendicular on P. Now, let us project the spheres on P. We get a set of circles with the property that any 3 of them intersect, and, thus, from Helly's theorem in a plane, we get that the circles intersect. Since the spheres intersect in an interior point, the circles have an actual surface in common, not only a point. Let A be a point in that surface. The line through A that is parallel to d is the required d'.

Of course, this problem isn't a hard topology problem, but it is pretty hard for the undergraduate level.

This article wasn't the different article that I promised. That one is still underway. Stay tuned folks, more problems/journals/randomness is coming.

Cris.