PreprintCC BY 4.0arXiv
Rectangular Pegs on Jordan Curves of Finite p-Variation
Abstract
We prove that every planar Jordan curve of finite p-variation, with 1≤ 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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BibTeX
@misc{li2026rectangular,
title = {{Rectangular Pegs on Jordan Curves of Finite $p$-Variation}},
author = {Xiangfei Li and Yichen Pan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04918},
eprint = {2609.04918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04918},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rectangular Pegs on Jordan Curves of Finite p-Variation
AU - Xiangfei Li
AU - Yichen Pan
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04918
UR - https://arxiv.org/abs/2609.04918
AB - We prove that every planar Jordan curve of finite p-variation, with 1≤ 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.
ER -