Concurrent normals to convex bodies and spaces of Morse functions
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 \( e: {\mathbb S}^{n-1} \to G_2 ({\mathbb S}^{n-1})\) assigns to a point \( x \in {\mathbb S}^{n-1}\) the function \( p \mapsto x \cdot p \). The author formulates and proves in the cases \( n = 2, 3, 4\) the following \textit{Conjecture 2}: \( G_{2n-1} ({\mathbb S}^{n-1})\) retracts to the image of \(e\). He shows also that Conjecture 2 implies a weak form of Conjecture 1. Reviewer's remark: Note that the text of the proof of Theorem 1 does not seem to be quite correct: ``Let \( V \subseteq \bar{K}\) be the set of points \( p \in \bar{K} \) such that \( d_p\) on \(\partial K \) has a unique local minimum. Similarly define \( W \subseteq \bar{K}\) to be the set of points where \( d_p\) on \(\partial K \) has a unique local maximum. It is easy to see that both \(V\) and \(W\) are closed.'' Simple examples of unit balls show that the sets \( V\), \(W\) do not need to be closed without additional assumptions. Actually, these examples do not contradict Theorem 1.
scientific article; zbMATH DE number 6005034