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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?

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    $\begingroup$ 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. $\endgroup$ – Chris Godsil Oct 17 '18 at 17:35
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    $\begingroup$ a sphere is an example of such metric spaces. $\endgroup$ – Mahdi Oct 17 '18 at 23:29

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