Rectangles inscribed in symmetric continua
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 \(Q\).
scientific article; zbMATH DE number 806266
scientific article