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If we want to talk about geometry in a metric space, we go through the following procedure to define geodesics. We define the length of a path in the (complete) space as the length given by integrating over infinitesimal distances on the path. The intrinsic metric of the space is then given by the infimum of the lengths of all paths between two points. If the intrinsic metric agrees with the metric we call the space a 'length space'; If furthermore there is a unique path between any two points whose length is the same as the distance between the points, we call that path the geodesic between the two points. We can from here go on and talk about geometry.

The above definition uses that property of straight lines in the euclidean plane, that they are the shortest paths between two points, to generalize to abitrary metric spaces. This is fine and also nicely connects with the usual definition of geodesics in Riemannian geometry, however it's only applicable in a restricted class of metric spaces (length spaces with unique geodesics), and seems to rely on further analytic properties instead of propeties related only to the metric itself.

However, here is an alternative defintion of straight lines in general metric spaces that seems to me much more general, applicable to many more metric spaces and also relies only on simple properties of of the metric. A further property of straight lines in the euclidean plane, besides the fact that they form the shortest path between two points, is that they are the locus of all points equidistant from two given points. Let us, then, define a straight line in a metric space as the locus of all points equidistant from two given points; I played with some equations and it seems to me that at least in taxicab space it gives the 'correct' lines. What is the problem with this definition? Why is it not used? Which properties of straight lines and the geometry are generalized to general metric spaces if we use this defintion, and which properties are lost?

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    $\begingroup$ In $\mathbb R^2$ with the $\ell^\infty$ norm (the taxicab space), it seems to me that the set of points equidistant from $(1,0)$ and $(-1,0)$ is $\{(0,y): y \in [-1,1]\} \cup \{(x,y): |y| > 1, x \in [-(|y|-1),|y|-1]\}$. $\endgroup$ Jan 27 '18 at 14:27
  • $\begingroup$ Actually, now that I think about it, $\ell^1$ is what is called the taxicab space. Note that $(\mathbb R^2,\ell^1)$ and $(\mathbb R^2,\ell^\infty)$ are isometric via the map $(x,y) \mapsto (x+y,x-y)$, so we have a similar example in $\ell^1$. The set of points equidistant from $(-1,-1)$ and $(1,1)$ in $\ell^1$ is the line between $(1,-1)$ and $(-1,1)$, together with the two cones with sides parallel to the axes and vertices at those points. $\endgroup$ Jan 27 '18 at 14:52
  • $\begingroup$ The notion of a line makes sense even for non-uniquely geodesic spaces. Also, in Riemannian geometry geodesics are only local length-minimizers. For instance, the equatorial circle on the round sphere is a geodesic. $\endgroup$ Jan 29 '18 at 2:20
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Even if you stick to the Euclidean metric, this is the wrong definition! It works for $\mathbb{R}^2$, but is horribly wrong in every other dimension. For instance, in $\mathbb{R}$ your "lines" would be single points, and in $\mathbb{R}^3$ your "lines" would be planes (in general, in $\mathbb{R}^n$ you would get $(n-1)$-dimensional affine subspaces). So, your definition may describe something interesting which is worth studying in general metric spaces, but it doesn't have much to do with "lines".

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