Let \(K\) be a symmetric (with respect to the origin) bounded convex body in \({\mathbb R}^n\), and \(N_K\) its Minkowski functional (or gauge). The bisector of the segment \([0,x]\) is the set of points which are equidistant of \(0\) and \(x\) for \(N_K\): \[ H_x=\{y\in {\mathbb R}^n ;\;N_K(y)=N_K(x-y)\} . \]
M. M. Day [Trans. Am. Math. Soc. 62, 320-337 (1947; Zbl 0034.21703)] stated that \(K\) is an ellipsoid if and only if all the bisectors are hyperplanes. It was pointed out that this follows immediately from a result of R. C. James [Duke Math. J. 12, 291-302 (1945; Zbl 0060.26202)].
In this paper, it is shown that \(H_x\) is a closed, connected set, with the following convexity property: whenever a line parallel to \(x\) intersects \(H_x\) in two distinct points, the corresponding segment is contained in \(H_x\). The main results of the paper are: For every strictly convex \(K\), the bisectors are all homeomorphic to a hyperplane (Theorem 2). The converse is not true, but, for \(n\geq 2\), if all the bisectors are homeomorphic to a hyperplane, then there is no \((n-1)\)-dimensional cylinder contained in the boundary of \(K\); furthermore, every maximal such cylinder with generators parallel to \(x\) (if any) is of dimension \((n-2)\) (Theorem 3). Several examples are given to illustrate this.
The paper ends with a result on Dirichlet-Voronoi cells in lattices.