Is the closure of a geodesically convex set convex?

Question

Is the closure of a geodesically convex set convex? If so, is there a simple proof for it? In $ \mathbb{R}^n $ there is a simple proof for it through convergent sequences. How should I apply it on this generalization?

Thanks in advance!

Answer 1

Maybe the following example fits you:

Consider $X := [0,1] \times [0,1[ ~\cup ~\{(0,1),(1,1)\} \subset \mathbb R^2$, equipped with the induced metric, and $Y := [0,1] \subset \mathbb R$, again equipped with the induced metric. Now form the topological space $Z = X \cup Y /\sim$ obtained by the disjoint union where the points $(0,1)$ and $0$, respectively $(1,1)$ and $1$, are identified. A metric on $Z$ is determined by

$d(p,q) := \text{ minimal length of a continuous curve in } Z \text{ from } p \text{ to } q,$

where length is defined in the obvious way here. Then the set $[0,1] \times [0,1[$, considered as a subset of $Z$, is geodesicaly convex but its closure (given by $X \subset Z$) is not.