The Rectangular Peg Problem

TL;DR

This paper proves that for any smooth Jordan curve and any rectangle, there exists a similar rectangle inscribed in the curve, using advanced topological and symplectic geometry techniques.

Contribution

It establishes the existence of inscribed similar rectangles in Jordan curves, extending classical geometric problems with novel topological methods.

Findings

Existence of similar rectangles inscribed in any smooth Jordan curve.

Application of Shevchishin's theorem to geometric problems.

Connection between symplectic topology and classical geometry.

Abstract

For every smooth Jordan curve $\gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $\gamma$. The proof relies on Shevchishin's theorem that the Klein bottle does not admit a smooth Lagrangian embedding in $\mathbb{C}^2$.