Metric characterizations of projective-metric spaces (Q6610769)

The author starts by recalling Hilbert's Problem IV of the list of problems he proposed in 1900 at the Paris International Congress of Mathematicians. The problem asks for the construction of all projective metrics on an open convex subset of projective space, that is, the metrics whose geodesics are the intersections of this open subset with the projective lines of the ambient space. The author addresses the general question of characterizing such spaces under the effect of adding some additional conditions on the metric. In doing this, he surveys notions like Hilbert and Minkowski metrics, projective center, Ptolemaic metric, Erdös ratio, conics in metric spaces, general Finsler projective metrics, the Ceva and Menelaus properties, and other properties of triangles in a projective-metric setting. Notions such as plane and line perpendicularity, equidistance of lines, bisectors, medians, bounded curvature and others, that hold in a general metric space, are used. These properties were introduced in such a general setting by Herbert Busemann. The questions of characterizing Minkowski planes, of Hilbert geometries and of the three classical geometries, which were extensively investigated by Busemann, are addressed by the author, who also mentions several open problems on this topic.\N\NFor the entire collection see [Zbl 1537.51001].