It is known that convexity implies contractibility in Euclidean spaces. I want to know whether it holds in a general length space. To be specific:
Let $(X,d)$ be a length space and $A\subseteq X$. $A$ is said to be convex if for any $x,y\in A$, there is a shortest curve in $A$ joining $x,y$. Is $A$ contractible or not?
The question arises when I am reading a classical paper by S. Alexander and R. Bishop. In this paper, convexity is defined as follows: $A$ is called convex if for any two scaled geodesics $\alpha,\beta:[0,1]\to A$ the function $d(\alpha(t),\beta(t))$ is convex. This is a condition similar but wearker than having nonpositive curvature. An argument in section 3 shows that such convex sets are contractible. However, this kind of convexity seems so 'strong' that I cannot come up with an example other than geodesic spaces with nonpositive curvature.
Thanks for your time and efforts!
Well, I'm still not sure what a length space is; there is at least one major problem with the definition given in a comment above. But, trying to extract the spirit of the problematic definition:
Say $S=\:z\in\Bbb C:|z|=1\}$, and define a metric on $S$ by saying $$d(e^{it},e^{i(t+\delta)})=|\delta|$$if $|\delta|\le\pi$. Is $(S,d)$ a length space? It's certainly not contractible.