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. Tao also formulated several related conjectures. Jordan curves close to a C2 Jordan curve In March 2022, Gregory R. Chambers showed that if γ is a Jordan curve which is close to a C2 Jordan curve β in R2, then γ contains an inscribed square. He showed that, if κ>0 is the maximum unsigned curvature of β and there is a map f from the image of γ to the image of β with ||f(x)−x|| < 1/10κ and f∘γ having winding number 1, then γ has an inscribed square of positive sidelength. Variants and generalizations One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle T and Jordan curve C, there is a triangle similar to T and inscribed in C. Moreover, the set of the vertices of such triangles is dense in C. In particular, there is always an inscribed equilateral triangle. 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 R3; his proof is published in Meyerson. In 2020, Morales and Villanueva characterized locally connected plane continua that admit at least one inscribed rectangle. In 2020, Joshua Evan Greene and Andrew Lobb proved that for every smooth Jordan curve and rectangle R in the Euclidean plane there exists a rectangle similar to R whose vertices lie on C. This generalizes both the existence of rectangles (of arbitrary shape) and the existence of squares on smooth curves, which has been known since the work of . 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 C in Rn satisfying "Condition A" that no two chords of C in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals. This class of curves includes all C2 curves. Nielsen and Wright proved that any symmetric continuum K in Rn contains many inscribed rectangles. References Further reading External links Mark J.