Description
Projective metrics have the property that the distances of any point of any linear segment from the endpoints sum up to a constant, the distance of the endpoints. The question arises if the quadratic curves has a similar property, namely, the distances of any point of a quadratic curve from two fixed points sum up to a constant, bigger than the distance of the fixed points. Such projective metrics are called quadratic, and it is my conjecture, that exactly the projective metrics of constant curvature are quadratic. Previously only Beltrami's theorem from 1865 and Busemann's theorem from 1953 supported the conjecture. These theorems state respectively that if a projective metric is Riemannian, i.e. every infinitesimal sphere is quadratic, than it is of constant curvature, and a sphere of a Minkowski-metric is quadratic if and only if the metric is Euclidean. The talk describes the background of the conjecture and presents some new supporting results of mine with some words about the proofs.