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I was discussing means with my friend, and I tried to illustrate the concept of geometric mean using the following idea:

Suppose we have two positive quantities $x,y>0$. The simplest geometric object we can make out of those is an $x \times y$ rectangle. What if we want a regular rectangle (i.e., a square) that "best approximates this rectangle"?

One possibility is a square of side length $$\ell_1 =\frac{x+y}{2} ,$$ keeping the perimeter the same at $2x+2y$. Another candidate is $$\ell_2 =\sqrt{xy} ,$$ this time keeping the area the same at $xy$.

I then realized I can generalize this idea to higher dimensions: If we have three positive numbers $x,y,z>0$, consider a $x \times y \times z$ rectangle, and a cube whose side $\ell$ is to be decided:

  • Keeping the 1-dimensional "length-of-the-skeleton" the same we get $$4x+4y+4z=12 \ell_1 \implies \ell_1=\frac{x+y+z}{3}. $$
  • Keeping the 2-dimensional area of the faces the same we get $$2xy+2xz+2yz=6\ell_2^2 \implies \ell_2=\sqrt{\frac{xy+xz+yz}{3}}.$$
  • Keeping the 3-dimensional volume the same we get $$x y z =\ell_3^3 \implies \ell_3=\sqrt[3]{x y z}.$$

Notice that among the usual arithmetic and geometric means, a different kind of mean has popped up.

This idea can go further, using "$n$-orthotopes" or hyperrectangles, producing $n$ distinct means from any sequence $x_1,\dots,x_n$ of positive quantities:

For $1 \leq d \leq n$ let $e_d(x_1,\dots, x_n)$ denote the elementary symmetric polynomial on $n$ symbols of degree $d$. We define $$\ell_d(x_1,\dots,x_n) := \sqrt[d]{\frac{e_d(x_1,\dots,x_n)}{\binom{n}{d}}}.$$

I have two questions about this:

  1. Is this concept already known?
  2. I believe that the AM-GM inequality generalizes to $\ell_1 \geq \ell_2 \geq \cdots \geq \ell_n$. Is this correct?

Thank you!

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Yes, that relationship between the elementary symmetric polynomials holds, it is known as Maclaurin's inequality, and a consequence of Newton's inequalities.

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