Okay, folks, time for another Problem of the Week! I hope it generates more response than last week’s problem:

Let be a simple closed curve in the plane, and let be any point strictly in the region interior to . Show there are two points on whose midpoint is .

Please submit solutions to **topological[dot]musings[At]gmail[dot]com** by **Wednesday, July 9**, 11:59 pm (UTC); do **not** submit solutions in Comments. Everyone with a correct solution will be inducted into our Hall of Fame! We look forward to your response.

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July 7, 2008 at 6:47 pm

Anonymouswhat does it mean by 2 points on C whose midpoint is P?

July 7, 2008 at 7:21 pm

PaulI think they mean that:

There exist points c1 and c2 on C such that the line segment joining c1 to c2 has midpoint p.

July 7, 2008 at 8:36 pm

Todd TrimblePaul is right. In other words, p is the average (c_1 + c_2)/2 in the sense of linear or (better yet) affine algebra.

July 8, 2008 at 2:47 am

Paul ShearerWhile trying to solve this, I came up with a strong conjecture that implies the desired statement. I think the conjecture is harder to prove than this problem is, so it should be no spoiler if I state it.

First, suppose p = 0 in the above problem without loss of generality by translating p to the origin.

Conjecture: The following set is connected:

S = { (p_1, p_2) \in C \cross C : p_1 = c*p_2 for some c \in (-\infty, 0) }.

The POW proposition is a (slightly nontrivial) corollary of this conjecture. Physically the conjecture corresponds to the following claim:

Suppose you have a rigid but extendable rod anchored at the origin and attached to the curve C (think of it as a steel track) at two points p_1 and _2 by rolling bearings. Thus, when you push the rod along the track C, p_1 and p_2 will move, the rod will expand and contract so that it stays on the track, but the rod always will pass through the origin. I claim that this rod can reach any possible configuration from any other possible configuration on C.

If you draw a few diagrams and watch the rod move, the claim is very compelling… but can it be proved? I think this claim sits at an interesting nexus of geometry and topology, and I’d like to see a solution.

July 10, 2008 at 2:31 am

Paul ShearerAfter some more thought, I’ve determined my conjecture is false. I sent Todd a counterexample where the curve C is a circle with two “handles” (think starship enterprise). If anyone cares I’ll send the details…

July 10, 2008 at 6:19 pm

Solution to POW-7: Midpoints between points on a curve « Todd and Vishal’s blog[...] Trimble The solutions are in! There was quite a bit of activity behind the scenes on this one; POW-7 might look intuitively obvious, as if it should succumb to an easy application of the intermediate [...]

July 22, 2008 at 7:10 am

Analyzing the hairy ball theorem « Todd and Vishal’s blog[...] of weeks ago, when Miodrag Milenkovic posed an interesting general problem in connection with POW-7, I was reminded of the “hairy ball theorem” (obviously a phrase invented during a more [...]