On bisectors in Minkowski normed spaces

We discuss the concept of the bisector of a segment in a Minkowski normed n-space, and prove that if the unit ball It of the space is strictly convex then all bisectors are topological images of a hyperplane of the embedding Euclidean n-space. The converse statement is not true. We give an example in the three-space showing that all bisectors are topological planes, however K contains segments on its boundary. Strict convexity ensures the normality of Dirichlet-Voronoi-type K-subdivison of any point lattice.