Minkowski spaces with a nontrivial group of isometries represent a nowhere dense set (Q914115)

Let \({\mathcal M}\) be the class of Minkowski spaces (finite-dimensional real normed spaces) with nontrivial (i.e. containing elements not being a translation) group of isometries. The author proves that \({\mathcal M}\) forms a nowhere dense class in the class of all Minkowski spaces, both in the case of the topology induced by the Hausdorff metric and in the case of the topology determined by the Banach-Mazur distance function. An easy proof shows that the class \({\mathcal M}\) is closed. Then it remains to prove that the class of Minkowski spaces with trivial isometry group is everywhere dense. This is a consequence of the Belenot's result; however in the finite-dimensional case the author gives another easy and elegant proof of this fact. The theorem proved in the note once more indicates how exceptional are geometries admitting nontrivial isometries.