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It is well-known that the geometric center is found by finding the arithmetic mean of all points but how is geometric center related to geometric mean? Or is there no relation except their names?

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If the geometric center of a finite collection of points in, say, ${\bf R}^2$ is their arithmetic mean, then it has no relation to the geometric mean. Indeed, it's not clear what the geometric mean means, since first you'd have to define how to multiply two points.

The geometric mean of a finite collection of positive numbers is less than or equal to the arithmetic mean.

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