I'm attending a lecture series about introduction to non-Euclidian Geometry, but it is focused on the intuition of that topic without giving me the tools to analize the following question:
Is the hyperbolic plane convex?
Where convex here means that there is geodesic line between any two points in the set.
At one side I feel like it should be convex, because $\mathbb{H}$ has no 'obstructions', but the 'abundante of space' makes me wonder if could be a straight line connecting any two points. I would be glad to see a proof of it.
Here's a tool you can use to answer this question: In the upper half plane model of the hyperbolic plane $$\mathbb H^2 = \{(x,y) \in \mathbb R^2 \mid y > 0\} $$ (with the metric $ds^2 = \frac{dx^2 + dy^2}{y^2}$), the geodesic lines are the Euclidean semicircles that hit the $x$-axis at right angles and the Euclidean vertical rays that are based at points on the $x$-axis.
Now convince yourself that for any two points $p,q \in \mathbb H^2$ there exists a geodesic line containing both $p$ and $q$, i.e. either a Euclidean semicircle through $p$ and $q$ hitting the $x$-axis at right angles or a Euclidean vertical ray through $p$ and $q$ based at a point on the $x$-axis.