Polygons inscribed in Jordan curves with prescribed edge ratios

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-05-29 DOI:10.1016/j.topol.2024.108971

Yaping Xu , Ze Zhou

引用次数: 0

Abstract

Let J be a simple closed curve in $R^{k}$ $(k \geq 2)$ that is differentiable with non-zero derivative at a point $A_{0} \in J$ . For a tuple of positive reals $a_{1}, \dots, a_{n}$ $(n \geq 3)$ , each of which is less than the sum of the others, we show that there exists a polygon $Q_{n}$ inscribed in J with sides of lengths proportional to $(a_{1}, \dots, a_{n})$ . As a consequence, we prove the existence of triangle inscribed in J similar to any given triangle.

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以规定边长比嵌入约旦曲线的多边形

设是一条简单的闭合曲线，在点处可微且导数不为零。对于每一个都小于其他两个之和的正整数元组，我们证明存在一个多边形，它的边长与 .因此，我们证明了存在一个与任意给定三角形相似的三角形。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.