Inscribed square problem

The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some square? This is true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. Some early positive results were obtained by Arnold Emch and Lev Schnirelmann. As of 2016^[update] the general case remains open.

Let C be a Jordan curve. A polygon P is inscribed in C if all vertices of P belong to C. The inscribed square problem asks:

It is not required that the vertices of the square appear along the curve in any particular order.

Some figures, such as circles and squares, admit infinitely many inscribed squares. If C is an obtuse triangle then it admits exactly one inscribed square; right triangles admit exactly two, and acute triangles admit exactly three.

It is tempting to attempt to solve the inscribed square problem by proving that a special class of well-behaved curves always contains an inscribed square, and then to approximate an arbitrary curve by a sequence of well-behaved curves and infer that there still exists an inscribed square as a limit of squares inscribed in the curves of the sequence. One reason this argument has not been carried out to completion is that the limit of a sequence of squares may be a single point rather than itself being a square. Nevertheless, many special cases of curves are now known to have an inscribed square.

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