Concurrent normals to convex bodies and spaces of Morse functions
type: Item
instance of: scholarly article
title: Concurrent normals to convex bodies and spaces of Morse functions
zbMATH Open document ID: 1242.52006
author: John Pardon
published in: Mathematische Annalen
publication date: 2012-02-10T00:00:00Z
review text: The work is motivated by the \textit{Conjecture 1}: ``If \(K \subset {\mathbb R}^n\) is a bounded open convex set, then there exists a point \( p \in K\) which is the point of concurrence of \(2n\) normals from points on \( \partial K \)'', formulated by \textit{H. T. Croft, K. J. Falconer} and \textit{R. K. Guy} [Unsolved problems in geometry. Problem Books in Mathematics 2. New York etc.: Springer-Verlag (1991; Zbl 0748.52001)]. Let \( d_p: \partial K \to {\mathbb R} \) (\( p \in K\)) be the distance squared to \(p\). Local extrema of \(d_p\) correspond to normals passing through \(p\). The paper is devoted to estimates of numbers of these normals in low dimensions. Lemma 1. If \(K \subset {\mathbb R}^2\) is a bounded open convex set, then there exists a point \( p \in K\) which is the point of concurrence of four normals from points on \( \partial K \). Theorem 1. If \(K \subset {\mathbb R}^3\) is a bounded open convex set, then there exists a point \( p \in K\) which is the point of concurrence of six normals from points on \( \partial K \). Theorem 2. If \(K \subset {\mathbb R}^4\) is a bounded open convex set whose boundary is of class \(C^{1,1}\), then there exists a point \( p \in K\) which is the point of concurrence of eight normals from points on \( \partial K \). If \( \partial K \) is of class \(C^1\), then either there exists a point \( p \in K\) where eight normals concur or there exists a point \( p \in \partial K\) where infinitely many normals concur. The proofs are based on considerations of homotopy properties of the following functional spaces: Let \( G^{\circ}_k ({\mathbb S}^{n-1}) \subset C^1({\mathbb S}^{n-1})\) be the set of functions \( f \) on \({\mathbb S}^{n-1}\) for which \( \nabla f = 0 \) at no more than \(k\) points. Let \( G_k ({\mathbb S}^{n-1}) \subset G^{\circ}_k ({\mathbb S}^{n-1}) \) be the closure in \( G^{\circ}_k \) of the set of Morse functions in \( G^{\circ}_k \), and let \( e: {\mathbb S}^{n-1} \to G_2 ({\mathbb S}^{n-1}…
reviewed by: Vladimir Golubyatnikov
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 6005034
zbMATH Keywords: Morse functions
zbMATH Keywords: points of concurrence
zbMATH Keywords: homotopy equivalence
zbMATH Keywords: retract
MaRDI profile type: Publication
full work available at URL: s00208-010-0629-9
OpenAlex ID: W2001890188
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DOI: 10.1007/S00208-010-0629-9
description: scientific article; zbMATH DE number 6005034