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Is there a named theorem that says for a k-dimensional hyperrectangle and a fixed sum $S$ of the side lengths in each dimension (what I called "components" in the title), the volume is maximal when the side lengths are equal (the object is a $k$-hypercube with side length $\frac{S}{k}$)?

(This implies that for a fixed perimeter, the $2D$ rectangle with he largest possible area is a square. But the above is just generalized to $k$-dimensions.)

It doesn't seem difficult to prove, but it seems fundamental enough to have a well-known theorem that one could cite if it were needed as part of another proof.

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Yes, this is a simple consequence of the inequality of arithmetic and geometric means

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