Rectangles inscribed in symmetric continua
type: Item
instance of: scholarly article
title: Rectangles inscribed in symmetric continua
zbMATH Open document ID: 0831.54032
DOI: 10.1007/BF01263570
author: Yanyan Li
published in: Geometriae Dedicata
publication date: 1996-02-13T00:00:00Z
review text: The authors deal with an old and difficult but very geometric problem, asking whether every simple closed curve on the plane admits an inscribed square. Stromquist has shown that every locally monotone simple closed curve on the plane admits an inscribed square, and Klee and Wagen have shown that every simple closed curve symmetric about a point \(z\), such that every line through \(z\) meets the curve in exactly two points, admits an inscribed square. Using basic observations, they prove that if \(J\) is a simple closed curve on the plane, which is either symmetric about a point \(z\) or symmetric through a line \(L\), then for any rectangle \(Q\) there is a rectangle similar to \(Q\) inscribed in \(J\). In the sequel, they prove the following more general results by using very delicate arguments: Theorem 1. Let \(K\) be a continuum in \(\mathbb{R}^n\) symmetric about a point \(z\) in its complement. Then there is a continuum of rectangles incribed in \(K\), each of which is symmetric about \(z\). Moreover, for any given rectangle \(Q\) there is a rectangle in this continuum similar to \(Q\). Theorem 2. Let \(K\) be a continuum in \(\mathbb{R}^n\) symmetric across a hyperplane \(H\). If \(K \cap H\) is disconnected, then there is a continuum of rectangles inscribed in \(K\), each of which is also symmetric across \(H\). Moreover, for any given rectangle \(Q\) there is a rectangle in this continuum similar to \(Q\).
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 806266
zbMATH Keywords: simple closed curve
zbMATH Keywords: inscribed square
reviewed by: Joachim Grispolakis
MaRDI profile type: Publication
Wikidata QID: Q61920152
cites work: Q4117061
cites work: Mountain Climbing, Ladder Moving, and the Ring-Width of a Polygon
cites work: Q4002287
cites work: Inscribed squares and square‐like quadrilaterals in closed curves
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description: scientific article; zbMATH DE number 806266