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In normal Euclidean space with the $L_2$ metric, the shortest path between two points is a straight and unique line. However, on the taxi-cab metric ($L_1$), between any two points that do not lie on the same vertical or horizontal line, there are an infinite number of shortest paths between all with the same path distance.

Is there a name for and/or a way determine whether a given metric has unique shortest paths?

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  • $\begingroup$ Roughly, a space with paths of minimal length between pairs of points is a length [metric] space. If the paths are unique, I would suggest that the phrase unique length [metric] space is likely unambiguous. $\endgroup$
    – Xander Henderson
    Mar 16 '18 at 15:53
  • $\begingroup$ As to the question of finding paths of minimal length, that is a more difficult problem (while I was typing, Simonsays gave one example). For example, the Sierpinski gasket is a length space, but contains pairs of points for which there are two paths of minimal length, and others for which the path of minimal length is unique. It is not obvious (though true) that two is the maximum number of minimal paths. More pathological examples exist. $\endgroup$
    – Xander Henderson
    Mar 16 '18 at 15:56
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you might be interested in Riemannian manifolds $(M,d)$, which are manifolds that come along with a metric induced by $d$.
This metric then helps to find (locally) shortest paths between points on that manifolds. These paths are called geodesics and are solutions of ODE's equations. In Euclidean (flat) spaces, they are just straight lines, but e.g. on a sphere, they are given by the great circles.
In other words, if you can give your object a smooth manifold structure, you can find unique shortest paths. Hope this helps.

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    $\begingroup$ Not all Riemannian manifolds have the property described in the question. For instance, on the usual sphere any two antipodal points may be joined by infinitely many paths of minimal length. $\endgroup$
    – user228113
    Mar 16 '18 at 16:48
  • $\begingroup$ yes you are right, this is why I mentioned its a local property $\endgroup$
    – Simonsays
    Mar 16 '18 at 17:08

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