Geodesics in a complete metric space

Question

Let $(X,d)$ be a complete metric space.

Is it always true that for every point $x\in U$ there exists an $r_x>0$ sufficiently small such that for every $y\in B_{r_x}(x)$ (the closed ball of center $x$ and radius $r_x$) there is a geodesic for $d$ from $x$ to $y$? Note that such geodesic doesn't need to lie inside of $B_{r_x}(x)$.

If the answer is "no", then are there some additional conditions on $(X,d)$ which will guarantee that the previous property is verified?

Answer 1

A geodesic exists iff a midpoint $z$ exists for any $x,y$. A point $z$ is defined as a midpoint if (def.) $d(x,z)=d(z,y)= \frac {1}{2}d(x,y)$

See Thm 1.6 in http://math.ucr.edu/~monnot/geometry.pdf