On the false pole problem
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\(\, K\).
scientific article; zbMATH DE number 5189626