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In geometry, if I calculate the mean of a set of points, I'll find the center. This is useful. What happens if I calculate the median and are the results useful?

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In geometry, if I calculate the mean of a set of points, I'll find the center. This is useful. What happens if I calculate the median and are the results useful?
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the median assumes that there is a ordering to all of it but there are various different ways one could order all 3D points leading to different medians.
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How do you even define the median of a set of points?

You could compute the medians in each coordinate, but I'm pretty sure that will be different in different coordinate systems, so it's not very geometrically meaningful.
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