TITLE: Name of a metric space concept
QUESTION [8 upvotes]: I have a metric space with the following property (a bit like having unique geodesics): for any points $a,b,x,y$ with $d(a,b)=d(a,x)+d(x,b)=d(a,y)+d(y,b)$ and $d(a,x)=d(a,y)$, we have $x=y$.  Is there an established name for this?
(UPDATE: the condition $d(a,x)=d(a,y)$ was omitted by mistake in the original question.)

REPLY [3 votes]: It is easy to prove that the completion of your space can be any separable metric space 
where metric spheres are nowhere dense.

Does not it scare you?

Answer to the comment:
Not all of your spaces can be embedded into a metric tree.
BTW, there is a nice characterization of subsets in a metric tree:
$$ | x - y | + | a - b | \le \max\{|x-a| + |y-b|,|x-b|+|y-a|\}$$
for all points $x,y,a,b$
(here $|x-y|$ denotes the distance from $x$ to $y$).
In other words, the values 
$$X=|x-a|+|y-b|,$$
$$Y=|x-b|+|y-a|,$$ 
$$Z=|x-y|+|a-b|$$ satisfy ulrtatriangle inequality.
This inequality  can be also thought as a discrete analog of CAT[−∞] inequality.