Conics in Minkowski geometries (Q1784267)

With conics defined in the usual way, the metric being that of the Minkowski geometry, as the locus of all points \(X\) whose distance to a fixed point is a constant multiple of the distance from \(X\) to a fixed hyperplane, the author proves several characterizations of Euclidean space among Minkowski geometry. Euclidean geometry is the only Minkowski geometry that has a centrally symmetric conic. Euclidean geometry is the only plane Minkowski geometry in which a quadratic conic (one which is of one of the following three types: (i) \(\{(x,y): x^2+y^2=1\}\), (ii) \(\{(x,y): x^2-y^2=1\}\), and (iii) \(\{(x,y): x=y^2\}\)) exists.