every local monotone plane simple curve admits an inscribed square. The condition for the admission to happen is that for any point , the curve should be locally represented as a graph of a function . In more precise terms, for any given point on , there is a neighborhood and a fixed direction (the direction of the “-axis”) such that no chord of -in this neighborhood- is parallel to . Locally monotone curves include all types of polygons, all closed convex curves, and all piecewise C1 curves without any cusps. Curves without special trapezoids An even weaker condition on the curve than local monotonicity is that, for some ε > 0, the curve does not have any inscribed special trapezoids of size ε. 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 C, as C may not be rectifiable. 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. Curves in annuli If a Jordan curve is inscribed in an annulus whose outer radius is at most times its inner radius, and it is drawn in such a way that it separates the inner circle of the annulus from the outer circle, then it contains an inscribed square. 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. The limit of a sequence of large squares must again be a large square, rather than a degenerate point, so the limiting argument may be used. Symmetric curves The affirmative answer is also known for centrally symmetric curves, even fractals such as the Koch snowflake, and curves with reflective symmetry across a line. Lipschitz graphs In 2017, Terence Tao published a proof of the existence of a square