有限p-变分的若尔当曲线上的矩形嵌入
Rectangular Pegs on Jordan Curves of Finite $p$-Variation
- Soochow Academy（苏州大学苏东书院）
- Qiuzhen College Tsinghua Univerisity（清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明1≤p<2的平面有限p-变分若尔当曲线内接任意相似类矩形，结合Asano和Ike准则与变分控制逼近完成证明，为这类曲线的矩形嵌入问题提供了严格结论。
AI中文摘要：
我们证明，对于1≤p<2的每一条平面有限p-变分若尔当曲线，都内接任意给定相似类的矩形，特别地，这类曲线都内接正方形。证明结合了Asano和Ike的最新准则与变分控制逼近论证。我们证明，对任意q>p，有限p-变分若尔当曲线可在q-变分拓扑中被光滑若尔当嵌入逼近，该构造使用Boedihardjo和Geng的简单多边形插值、变分半范数间的初等插值不等式，以及多边形嵌入的变分控制平滑，再通过杨积分得到对应原函数的局部一致收敛。
英文摘要：
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.
