Let $(X,d)$ be a complete metric space with this property:
for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$.
I want to know if the family of complete metric spaces with this property are known or some work have been done on it.
Remark: I studied about metrically convex space which had been introduced by K.Menger(1928) and I am not sure if there is a relation between this family with my mentioned metric spaces.
