Inscribed square problem
This is true if the curve is convex or piecewise smooth and in other special cases.
[1] Some early positive results were obtained by Arnold Emch[2] and Lev Schnirelmann.
[5] 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.
Nevertheless, many special cases of curves are now known to have an inscribed square.
[6] Arnold Emch (1916) showed that piecewise analytic curves always have inscribed squares.
Emch's proof considers the curves traced out by the midpoints of secant line segments to the curve, parallel to a given line.
Therefore, there always exists at least one crossing, which forms the center of a rhombus inscribed in the given curve.
By rotating the two perpendicular lines continuously through a right angle, and applying the intermediate value theorem, he shows that at least one of these rhombi is a square.
[6] Stromquist has proved that every local monotone plane simple curve admits an inscribed square.
[7] The condition for the admission to happen is that for any point p, the curve C should be locally represented as a graph of a function
An even weaker condition on the curve than local monotonicity is that, for some
, the curve does not have any inscribed special trapezoids of size
A special trapezoid is an isosceles trapezoid with three equal sides, each longer than the fourth side, inscribed in the curve with a vertex ordering consistent with the clockwise ordering of the curve itself.
Its size is the length of the part of the curve that extends around the three equal sides.
Here, this length is measured in the domain of a fixed parametrization of
Instead of a limit argument, the proof is based on relative obstruction theory.
This condition is open and dense in the space of all Jordan curves with respect to the compact-open topology.
In this sense, the inscribed square problem is solved for generic curves.
[6] If a Jordan curve is inscribed in an annulus whose outer radius is at most
In this case, if the given curve is approximated by some well-behaved curve, then any large squares that contain the center of the annulus and are inscribed in the approximation are topologically separated from smaller inscribed squares that do not contain the center.
[8] In 2017, Terence Tao published a proof of the existence of a square in curves formed by the union of the graphs of two functions, both of which have the same value at the endpoints of the curves and both of which obey a Lipschitz continuity condition with Lipschitz constant less than one.
[9] In 2024, Joshua Greene and Andrew Lobb published a preprint improving this result to curves with Lipschitz constant less than
[11] One may ask whether other shapes can be inscribed into an arbitrary Jordan curve.
It is also known that any Jordan curve admits an inscribed rectangle.
This was proved by Vaughan by reducing the problem to the non-embeddability of the projective plane in
[15] In 2020, Morales and Villanueva characterized locally connected plane continua that admit at least one inscribed rectangle.
[16] In 2020, Joshua Evan Greene and Andrew Lobb proved that for every smooth Jordan curve
[3] In 2021, Greene and Lobb extended their 2020 result and proved that every smooth Jordan curve inscribes every cyclic quadrilateral (modulo an orientation-preserving similarity).
[19] Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general continua in higher dimensional Euclidean spaces.
For example, Stromquist proved that every continuous closed curve
