Characterization of linearity in terms of metric At least in Euclidean geometry and the upper half plane model of hyperbolic geometry, the statements '$y$ lies on the line segment determined by $x$ and $z$ ' and '$d(x,y)+d(y,z)=d(x,z) $' are equivalent.
I wonder whether this characterization holds in other geometries and whether it has a name to it. Do geometries with this relation have some special properties? 
I also think it is somehow related to the notion of uniform convexity in Banach spaces, but that is a different problem since we don't ask for vector space sturcture here.
Any insight or reference would be helpful!
Thanks!
 A: You don't ask for vector space structure, but you do want to have a notion of a line segment. What are the spaces where we have the latter without the former? Well, the projective space $\mathbb R P^n$ is an example of such space. And of course we can also  consider its convex subsets, where any two points can be connected by a lone segment. 
The investigation of metrics that satisfy the property you describe can be understood as a version of Hilbert's 4th problem, though Hilbert's meaning is not very clear. I know two books on this subject: Projective geometry and projective metrics by Busemann and Kelly, and Hilbert's fourth problem by Pogorelov. Busemann/Kelly call such metrics projective while Pogorelov calls them Desarguesian.
This EOM page uses the name "Hilbert geometry". I don't like this usage, because in the literature I know "Hilbert geometries" are those that use cross-ratio of four points to define the metric (google it). This is a special (very interesting) case of projective/Desarguesian metrics.
