Inscribed squares in plane curves

Abstract

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by x = xit), y = yit).These are real analytic periodic functions with period T. In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares.This theorem is similar to the theorem of Kakutani [l] that there exists a circumscribing cube around any closed, bounded convex set in E3.The latter theorem has been generalized by Yamabe and Yujobo [2], and Cairns [3] to show that in En there are families of such cubes.Here, for the case of squares inscribed in plane curves, we remove the restriction to convexity and give certain other results.A square inscribed in a curve C means a square with its four corner points on the curve, though it may not lie entirely in the interior of C. Indeed, the