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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!

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A Banach space with this property is called rotund, which means that the unit sphere contains no straight line segments, or equivalently every point of the unit sphere is an extreme point. I think you will find this is a weaker property than uniform convexity. – Rob Arthan Feb 25 '12 at 10:24
@martin. take the sup norm on $\mathbb{R}^2$, so $\|(x, y)\|$ is the larger of $|x|$ and $|y|$. Then equality holds in the triangle inequality for $x = (0, 0)$, $y = (1, a)$ and $z = (2, 0)$ for any $a$ with $-1 \le a \le 1$, but $y$ is only on the line segment $[x, z]$ if $a = 0$. – Rob Arthan Feb 25 '12 at 17:03
@Rob: my bad. I was thinking only of the implication $\Rightarrow$ – Martin Argerami Feb 25 '12 at 17:28

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.

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