Rectangular Pegs on Jordan Curves of Finite $p$-Variation

Xiangfei Li; Yichen Pan

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arXiv · 2609.04918

Rectangular Pegs on Jordan Curves of Finite $p$-Variation

Xiangfei Li·Yichen Pan

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Abstract

We prove that every planar Jordan curve of finite $p$-variation, with $1\leq p<2$, inscribes a rectangle of every prescribed similarity class. In particular, every such curve inscribes a square. The proof combines the recent criterion of Asano and Ike with a variation-controlled approximation argument. We show that for every $q>p$, a Jordan curve of finite $p$-variation can be approximated in the $q$-variation topology by smooth Jordan embeddings. The construction uses simple polygonal interpolants of Boedihardjo and Geng, an elementary interpolation inequality between variation seminorms, and a variation-controlled smoothing of polygonal embeddings. Young integration then gives locally uniform convergence of the associated primitives.

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Xiangfei Li, Yichen Pan. 2026-09-04. Rectangular Pegs on Jordan Curves of Finite $p$-Variation. https://arxiv.org/abs/2609.04918

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