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

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.