# Weak convexity in graphs

I asked the following question in MathStackExchange, but I did not get any answer, and I think that this might be the appropriate venue for the question.

As we know, a finite undirected graph induces a metric space on the set $$V$$ of its vertices. A convex set of vertices is defined as a set $$S \subseteq V$$ such that for any $$u,v \in S$$, all shortest paths (or geodesics) joining $$u$$ and $$v$$ are contained in $$S$$. Now I wonder if anybody has studied a relaxation of this definition, that requires that at least one shortest path joining $$u$$ and $$v$$ be contained in $$S$$. In Euclidean space both definitions are obviously equivalent, since for every pair of points $$x,y$$ there is a unique geodesic joining them. Are there other metric spaces, besides graphs, where two points may be joined by more than one geodesic?

• A subgraph is often called_geodetic_ if the distance between any two vertices in the subgraph is equal to the distance in the larger graph, so you are looking for information about geodetic subgraphs. – Chris Godsil Oct 17 '18 at 17:35
• a sphere is an example of such metric spaces. – Mahdi Oct 17 '18 at 23:29