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Given a set of $3$ positive real numbers $\{a, b, c \}$, one way of interpreting the geometric mean is that it is the answer to the following question:

Find the edge length $\mu_3$ of a cube whose volume is equal to that of a rectangular solid with edge lengths $a, b,$ and $c$.

(Here the subscript "$3$" indicates that we are trying to equalize volume, a 3-dimensional measure.) Similarly one way of interpreting the arithmetic mean is that it is the answer to the following question:

Find the edge length $\mu_1$ of a cube whose total linear measure (i.e. the sum of the lengths of all edges) is equal to that of a rectangular solid with edge lengths $a, b,$ and $c$.

As soon as one notices these geometric interpretations, the following immediately suggests itself as an obvious related question:

Find the edge length $\mu_2$ of a cube whose surface area is equal to that of a rectangular solid with edge lengths $a, b,$ and $c$.

Solving this problem, one finds that $$\mu_2 = \sqrt{ \frac{ab+ac+bc}{3}}$$ a formula which is a kind of hybrid of the formulas for the AM and GM.

Generalizing, for a set of $n$ positive real numbers, one can define $n$ different types of "mean", each answering a question of the following type:

Find the edge length $\mu_k$ of an $n$-dimensional hypercube whose $k$-dimensional measure (i.e. the sum of the measures of the $k$-dimensional subcubes) is equal to that of the hypersolid whose edge lengths are $a_1, a_2, \ldots a_3$.

So for example, with $n=4$ there are four different "means" of the set $\{a,b,c,d\}$:

  • $\mu_1(a,b,c,d)=\frac{a+b+c+d}{4}$ is just the arithmetic mean of the four numbers.
  • $\mu_2(a,b,c,d) = \sqrt{ \frac{ab + ac + ad + bc + bd + cd}{6} }$ equalizes the $2$-measure
  • $\mu_3(a,b,c,d) = \sqrt[3] {\frac {abc + abd + acd + bcd}{4} }$ equalizes the $3$-measure
  • $\mu_4(a,b,c,d) = \sqrt[4] {abcd}$ is just the geometric mean of the four numbers.

In general, $$\mu_k(a_1,a_2,\ldots a_n) = \sqrt[k]{\frac{\sum \text{all products of } k \text{ members of the set} \{a_1, a_2, \ldots a_n\} }{C_{n,k}}}$$

Okay, my questions:

  1. Do these generalized means have a name? What (if anything) is known about them?
  2. In particular it seems to be the case that for any set of numbers, $\mu_1 \geq \mu_2 \geq \mu_3 \geq \cdots \geq \mu_n$, a generalization of the standard $AM-GM$ inequality. But this seems much harder to prove. Does anybody see a way to prove this?

Please feel free to edit this question (especially its tags; I'm not happy with the ones I am using) for clarity.

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See The Super-Symmetric Mean and Inequalities for Symmetric Means

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  • $\begingroup$ Thanks for those links. So, chasing them down, it turns out the means I asked about are generally known as the (nth roots of the ) elementary symmetric means, and the chain of inequalities I listed are known as the Maclaurin's Inequalities. Link at en.wikipedia.org/wiki/Maclaurin_inequality for those who are interested. $\endgroup$ – mweiss Jun 24 '14 at 21:21

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