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Given a list of $n$ nonnegative real numbers $a_1, a_2, \dots, a_n$, the geometric mean of that list is defined to be $$\sqrt[n]{a_1a_2\cdots a_n}.$$

In the case of $n=2$, there are a few standard "geometric" interpretations of the geometric mean (generally involving power of a point, to visualize how the length $\sqrt{a_1a_2}$ depends on the lengths $a_1$ and $a_2$).

For the case of $n\ge 3$, are there are also nice geometric ways of understanding what the geometric mean measures? If so, what are they?

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It's the side of an $n-$cube whose $n-$dimensional volume is equal to that of an $n-$box with sides $a_1,,\dots,a_n$

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