What are some interesting real-world applications of metrics? I'm looking for ideas of real-world applications of different types of metrics/distances — especially (but not only) taxicab and railway metrics. By "real-world" I mean something that directly impacts our lives, i.e. not mathematical applications such as "solving systems of linear equations".
Examples:

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*Taxicab metric helps first responders assess how long it takes to reach the destination.

*Hamming distance helps find genetic mutations.

Thank you!
 A: Different metrics such as Manhattan distance (taxicab metric) can help in implementing heuristics for AI. A simple example might be as follows:
In an 8-puzzle (n-puzzle) using the taxicab metric to implement a heuristic which would make AI decide which move is better to play by measuring the Manhattan distances of tiles to where they are supposed to be.
A: Beside the ones you mention, a few quick ones that come to mind at the moment:
Euclidean metric: usual real-world notion of spatial distance
Discrete metric: captures whether you are in the same location as another or not
Wasserstein metric (earth mover's distance): the cheapest way to transform one pile of dirt into another, where cost of moving dirt is amount of dirt to be moved times distance to be moved. It is a way to quantify distances between probability distributions and as such, has several applications (see here for applications in machine learning).
Hausdorff distance: the distance it would take me to travel between two sets in a zero-sum game I play with an adversary, where my adversary moves first and chooses my origin in one of the two sets to maximize my distance travelled and I move second and choose my destination in the other set to minimize my distance travelled. It is a way to quantify distances between sets and has applications in computer vision/computer graphics.
A: *

*Does GPS count? The system calculates distance by including space-time curvature, and thus by tweaking Euclidean distance. This is one of the most important every day application of Einstein's General Relativity.

*Neural networks. Metrics are of course deeply hidden in the theory. By the universal approximation theorem every continuous function can be approximated by a neural network with arbitrary precision. Meaning for every continuous function there's a neural network arbitrarily close to it. This has deep implications in the digital world, where nowadays neural networks are almost everywhere, even inside CPUs.

