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I've been looking into the Toeplitz' Conjecture and became very interested, so I began to study it.

Here is the conjecture:

For any Jordan curve $\space \gamma \space$, there exist four distinct points on $ \space \gamma \space$ such that these four points are the vertices of a square.

In order to study this, a mathematician H. Vaughan wrote a paper on the proof that:

For any Jordan curve $\space \gamma \space$, there exist four distinct points on $ \space \gamma \space$ such that these four points are the vertices of a rectangle.

But I can't seem to find the paper, or at least anyone else rigorously explaining the proof.

The best I've found is this video:

https://www.youtube.com/watch?v=AmgkSdhK4K8&t=313s

Though I would love a read on the proof.

Thank you.

Max Echendu
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2 Answers2

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Actually,I think this video gives a proof of the inscribed rectangle problem. The last step, which said that it's impossible to embed a Möbius Strip in the upper half space with boundary on the curve in the equatorial plan is justified by the following :

If we had one, we could paste the boundary of a disk (the disk is embedded in the lower half space) on the curve and then obtain and embedding of $\Bbb{RP}^2$ (projective plane) in $\Bbb{R}^3$.

But it's impossible : By Alexander duality if $X$ embeds in $\Bbb{R}^3$, then $H_1(X)$ has no torsion (Hatcher, Corollary 3.45).

Does it sound good ? Sorry for my English.

Rob Arthan
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Ploopy
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Vaughan's proof is summarized on page 71 of Mark D. Meyerson's article "Balancing acts", Topology Proc. 6 (1981), no. 1, 59–75. In his article, Meyerson cites:

[Va] H. Vaughan, Rectangles and simple closed curves, Lecture, Univ. of Ill. at Urbana, 1977(?).

This may have been the first time Vaughan's proof appeared in an article, but I am not certain.

P.S.: To some people, the "Inscribed Rectangle Problem" refers to the still open question, whether for any aspect ratio $r$, any smooth Jordan curve inscribes a rectangle with this aspect ratio.

Benjamin Matschke
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