Geodesic convexity is a concept that arises in the context of Riemannian geometry and more generally in the study of metric spaces. A set is termed geodesically convex if, for any two points within the set, the shortest path (geodesic) connecting these two points lies entirely within the set.
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A subset of a Riemannian manifold is geodesically convex when any two of its points can be joined by a minimizing geodesic segment lying in the subset. In the hyperbolic plane the joining geodesic segment is unique. Thus in the Beltrami-Klein model, geodesically convex subsets correspond to Euclidean convex sets.