On the Square Peg Problem and Its Relatives

TL;DR

This paper advances the Square Peg Problem by proving it for several open and dense classes of curves, extending known results, and explores related inscribed shape problems and topological counterexamples.

Contribution

It proves the Square Peg Problem for new classes of open curves, including dense ones, and investigates inscribed rectangles and a topological counterexample for octahedra on spheres.

Findings

Proved the Square Peg Problem for several open classes of curves.

Extended the problem's validity to dense classes of curves.

Identified a topological counterexample for inscribing octahedra on spheres.

Abstract

Toeplitz's Square Peg Problem asks whether every continuous simple closed curve in the plane contains the four vertices of a square. It has been proved for various classes of sufficiently smooth curves, some of which are dense, none of which are open. In this paper we prove it for several open classes of curves, one of which is also dense. This can be interpreted in saying that the Square Peg Problem is solved for generic curves. The latter class contains all previously known classes for which the Square Peg Problem has been proved in the affirmative. [footnote] We also prove results about rectangles inscribed in immersed curves. Finally, we show that the problem of finding a regular octahedron on metric 2-spheres has a "topological counter-example", that is, a certain test map with boundary condition exists.