# Generalized AM-GM Inequality

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 $$\ell_d(x_1,\dots,x_n) := \sqrt[d]{\frac{e_d(x_1,\dots,x_n)}{\binom{n}{d}}}.$$

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