While thinking about geodesic lines I started exploring subsets of a metric space that have the following property.
$ \forall a,b,c \in L, d(a,c)> d(a,b) \land d(a,c) > d(b,c) \implies d(a,c) = d(a,b) + d(b,c) $
Where L is the subset I'm investigating. In $\mathbb{R}^n$ with a Pythagorean metric straight lines have this property. Is there a name for this property?
In a metric space $(X,d)$, you define the metric interval between $x,y\in X$ as $$I(x,y) = \{z \in X \mid d(x,y) = d(x,z) + d(z,y)\}$$ Note that this metric interval is always closed (think sequences and continuity of $d$). With this notion, you can start to define and study things like convex and hyperconvex metric spaces, etc, despite not having an addition operation defined on $X$. I first saw this in a paper named "Gluing Hyperconvex Metric Spaces", by Benjamin Miesch. I think the paper can be found in arxiv. Maybe you can start looking out for more things there.
