Bisectors in Minkowski 3-space
A 0-symmetric, bounded, convex body \(K\) in Euclidean \(n\)-space \(E^n\) defines a norm whose unit ball is \(K\). It was shown by the author [Acta Math. Hung. 89, 233--246 (2000; Zbl 0973.52001)] that if \(K\) is strictly convex then every bisector \(H_{\mathbf x}\), that is, the collection of all points that are equidistant from 0 and \(x\), is a topological hyperplane. The converse is not true however. In the paper under review the author conjectures a characterisation of topological hyperplanes in terms of shadow boundaries: The bisectors are topological hyperplanes if and only if the corresponding shadow boundaries are \((n-2)\)-dimensional topological spheres. The author proves this conjecture for the 3-dimensional case. In section 2 the author examines the topological properties of shadow boundaries, providing two concrete examples that show that they need neither be arcwise connected nor manifolds. He also introduces general parameter spheres as a tool for a potential proof of his conjecture. The shadow boundary \(S(K,{\mathbf x})\) of \(K\) in direction \({\mathbf x}\) in the unit sphere \(S^{n-1}\) of \(E^n\) consists of all points \(P\) in the boundary \(bdK\) of \(K\) such that the line through \(P\) in direction \({\mathbf x}\) does not meet the interior of \(K\). The general parameter sphere of \(bdK\) in direction \({\mathbf x}\) is \(\gamma_\lambda(K,{\mathbf x})=\frac{1}{\lambda}[bd(\lambda K)\cap bd(\lambda K+{\mathbf x})]\) where \(\lambda\geq \lambda_0\) and \(\lambda_0\) is the smallest value \(t\) for which \(tK\) and \(tK+{\mathbf x}\) intersect. Each bisector \(H_{\mathbf x}\) is the union over all \(\gamma_{\lambda}(K,{\mathbf x})\) for \(\lambda\geq \lambda_0\). The author shows that the general parameter spheres provide a natural parametrization for the points in the positive part of \(bdK\) minus \(\gamma_{\lambda_0}(K,{\mathbf x})\) by a point of \(S^{n-2}\) and a parameter \(\lambda >\lambda_0\). Furthermore, \(S(K,{\mathbf x})\) is the limit of the \(\gamma_\lambda(K,{\mathbf x})\) with respect to the Hausdorff metric when \(\lambda\) tends to infinity. The last section 3 deals with the 3-dimensional case only and proves the conjecture in this case. Using the Schoenflies-Swingle theorem it is shown that the shadow boundary and parameter spheres for \(\lambda>\lambda_0\) are topological circles assuming that bisectors are topological planes in \(E^3\).
scientific article; zbMATH DE number 2096243