It seems that I have the needed example, but I want it to be simple and self-explaining...



> 
> Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing geodesics.
> 
> 
> 


**Definitions:**


* A metric $d$ is called *intrinsic* if for any two points $x$, $y$ and any $\epsilon>0$ there is an $\epsilon$-midpoint $z$; i.e. $d(x,z),d(z,y)<\tfrac12 d(x,y)+\epsilon$.
* A minimizing geodesic is *nontrivial* if it connects two distinct points.
* A meric space is *nontrivial* if it contains two distinct points.


**Comments:**


* Clearly, $X$ can not be locally compact.



Intrinsic metric with no geodesics
Well, the unit ball in $c\_0$ is almost what you want (there is no unique shortest curve between points). All we need now is to enhance "bypasses" and to give disadvantage to "straight lines". This can easily be done by taking the distance element to be $(2+\sum\_n 2^{-n}x\_n)^{-1}\|dx\|\_\infty$, which is never less than the usual distance element in $c\_0$ and never greater than 3 times it in the unit ball. Now, if we have any continuous finite length curve $x(t)$ from $y$ to $z$ parametrized by the arclength, we can easily shorten it by replacing the $m$-th position by the maximum of the actual value of $x\_m(t)$ and $y\_m+t(z\_m-y\_m)/d+\frac 12 \min(t,d-t)$, where $d$ is the length of $x(t)$, which will work if $m$ is large enough since $\max\_t|x\_m(t)|\to 0$ as $m\to\infty$ and both functions change slower than the distance along the original curve.


This is certainly self-explaining (the shortest curve escapes from $c\_0$ to $\ell^\infty$) but I do not know if it is simple enough for your purposes. 


