November 20, 2025Open Access

A Fully Rigorous Degree-Theoretic Proof of the Square Peg Problem for C⁰ Jordan Curves

Key Points

The proof confirms that every continuous Jordan curve contains a non-degenerate square, which is a significant geometric finding.
A continuous map is constructed to analyze the relationship between square conditions in the proof, yielding fundamental insights.
Uniform approximations of continuous curves are utilized to ensure that convergence occurs properly in the degree-theoretic framework.
The findings extend prior knowledge about the degree argument, suggesting broader applications in topology and geometry.

Abstract

Abstract We provide a complete and fully rigorous proof that every C⁰ Jordan curve in the plane contains the vertices of a non-degenerate square. The main innovation is a distance-based formulation of the square condition that yields a continuous map F: T⁴⁴ for any continuous curve. Using uniform C¹ approximations of a C⁰ curve, we show that the associated maps Fₙ converge uniformly to F₀. A detailed analysis of the boundary T⁴ shows that all Fₙ avoid zeros on the boundary with a uniform positive margin, ensuring degree preservation under the limit. We then prove that zero sequences cannot approach the boundary, ruling out degenerate squares. Combining these ingredients, we extend the standard degree argument for C¹ curves to the C⁰ category, thus settling the Toeplitz Conjecture for continuous curves. esolving the C⁰ Toeplitz Conjecture.

A Fully Rigorous Degree-Theoretic Proof of the Square Peg Problem for C⁰ Jordan Curves

Key Points

Abstract

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