On the false pole problem
type: Item
instance of: scholarly article
title: On the false pole problem
zbMATH Open document ID: 1126.52007
author: E. Morales-Amaya
author: D. G. Larman
published in: Monatshefte für Mathematik
publication date: 2007-09-11T00:00:00Z
review text: Let \(\mathbb E ^d\) and \(\mathbb P ^d\) be the \(d\)-dimensional Euclidean and respectively projective space, and \(K\subset \mathbb E^d\), \(d\geq 2\), a convex body. A generalization of the classical duality pole-polar is as follows. A point \(p\in \mathbb P ^d\) is said to be a false pole of \(K\) with respect to some hyperplane \(H\) if \(p\) is not a pole of \(K\) with respect to \(H\) and for every 2-plane \(\Pi\) passing through \(p\), the section \(\Pi \cap K\) has a pole whose corresponding polar is \(\Pi \cap H\). It is easily seen that if \(E\subset \mathbb E ^d\), \(d\geq 3\), is an ellipsoid and \(p\in \mathbb E ^d\) is a pole of \(E\) with polar \(Q_p\), then, for every hyperplane \(H\neq Q_p\), \(p\) is a false pole of \(E\) with respect to \(H\). The paper under review contains a proof of the fact that a convex body \(K\subset \mathbb E ^d\), \(d\geq 3\), having an interior false pole \(p\in \mathbb E ^d \setminus H\) with respect to \(H\) is necessarily an ellipsoid. An example shows that the approach undertaken by the authors can not be used to obtain the same conclusion if \(p\) is a false pole not belonging to int\(\, K\).
reviewed by: Mihai Cipu
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 5189626
zbMATH Keywords: ellipsoid
zbMATH Keywords: center of symmetry
zbMATH Keywords: hyperplane of symmetry
zbMATH Keywords: projective center of symmetry
zbMATH Keywords: projective hyperplane of symmetry
zbMATH Keywords: pole
zbMATH Keywords: polar
zbMATH Keywords: false center
zbMATH Keywords: false pole
MaRDI profile type: Publication
full work available at URL: s00605-007-0448-6
OpenAlex ID: W2059857210
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DOI: 10.1007/S00605-007-0448-6
description: scientific article; zbMATH DE number 5189626