Geodesic Metric Space

A geodesic metric space is a metric space in which any two points are joined by a path isometric to the real interval (Bridson and Haefliger 1999). Such a path realizes the distance between its endpoints. This condition requires a globally distance-minimizing path, beyond local minimization along a curve.

For example, Euclidean space and a connected graph with each edge realized as an interval of length 1, equipped with the path metric, are geodesic metric spaces. The CAT(0) comparison condition further controls distances between points on these paths.

See also

Alexandrov Space, CAT(0) Space, Distance, Geodesic, Length Metric, Length Space, Metric Space, RCD Space

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References

Bridson, M. R. and Haefliger, A. Metric Spaces of Non-Positive Curvature. Berlin, Germany: Springer-Verlag, 1999. https://doi.org/10.1007/978-3-662-12494-9.

Cite this as:

Weisstein, Eric W. "Geodesic Metric Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeodesicMetricSpace.html

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