Metric Characterizations of Projective-Metric Spaces

This chapter is concerned with the study of projective-metric spaces, that is, metrics on open subsets of projective space whose geodesics are the intersection of this open set with the lines of the ambient space. The stress is on the effect of additional conditions on these so-called “projective-metric spaces”, which lead to some characterization of special geometries. We rely heavily on the work of Herbert Busemann in this domain. We formulate many open problems on this subject.