Open Problem Garden - Inscribed Square Problem - Comments

Already solved (re: Inscribed Square Problem)

Anonymous — Thu, 25 Feb 2021 17:01:43 +0100

This was proven in https://arxiv.org/pdf/2005.09193.pdf

in all simple closed curves there are 4n points (re: Inscribed Square Problem)

Anonymous — Mon, 08 Nov 2010 00:50:46 +0100

Would you clarify? An obtuse triangle has only one inscribed square, so this theorem is not true for n>=2. Do you have a reference to this theorem? Strashimir Popvassilev

Yes (re: Inscribed Square Problem)

Anonymous — Mon, 07 Jun 2010 11:16:26 +0200

Yes. Walter Stromquist, Inscribed squares and square-like quadrilaterals in closed curves, Mathematika 36: 187-197 (1989).

This is flawed (re: Inscribed Square Problem)

Anonymous — Mon, 07 Jun 2010 11:14:05 +0200

The approximation argument is flawed: the squares on approximating curves may have sides decreasing to 0, in which case the limiting "square" degenerates to a point. In fact, Stromquist's theorem covers a much wider class of curves than C^1, but not all continuous curves.

no. (re: Inscribed Square Problem)

Anonymous — Wed, 02 Jun 2010 20:43:18 +0200

If it were true for C^1 curves, then since a Jordan curve is compact, it may be weierstrass approximated by a series of C^1 curves (indeed by curves whose component functions are polynomials) such that the series converges uniformly to the given jordan curve. Then by assumption, each curve in the sequence contains 4 points forming a square, and the sequence of squares can be regarded as (eventually) a sequence in the (sequentially) compact space of the 4-fold product of any closed epsilon enlargement of the area bounded by the original jordan curve. It follows that the sequence of squares contains a convergent subsequence, which can be shown to be a square lying on the original jordan curve.

Thus, proving the C^1 case proves the general case.

inscription of squares in simple closed curves (re: Inscribed Square Problem)

Anonymous — Tue, 02 Mar 2010 23:14:12 +0100

There is a theorem that says:

in all simple closed curves there are 4n points that are vertex of n squares (inf = > n > =1)

Jorge Pasin.

Is the conjecture known to (re: Inscribed Square Problem)

Anonymous — Wed, 25 Nov 2009 18:45:23 +0100

Is the conjecture known to be true for C^1-smooth curves?

Quantifier (re: Inscribed Square Problem)

Anonymous — Mon, 28 Apr 2008 15:53:20 +0200

Phrasing should be changed from "Does any..." to "Does every..."

Inscribed Square Problem

dlh12 — Thu, 10 Apr 2008 23:37:02 +0200

Author(s): Toeplitz

Subject: Topology

Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?