I hope you can help me finding out if what I guess is right...
I consider a geodesic space X and two points x,y in this space, and I consider X with the property that, by considering its subspace G(x,y) containing all the geodesics between x and y, this G(x,y) is simply connected for all couples of points in X.
By trying to find some examples, I have started thinking about the following, is it true that for all a,b in G(x,y), G(a,b) is contained in G(x,y). I can't either prove this or find a counterexample.
I have tried to prove it in a lot of different ways but somehow I am not finding the right one and now I have started thinking that maybe it is wrong, but in this case I would like to find a counterexample, which I am not able to find!
Thanks for your help and suggestions!