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Here's a proof of the inequality of the arithmetic and geometric means in the form $$\frac{x_1^n}{n} + \cdots + \frac{x_n^n}{n} \geq x_1\cdots x_n.$$

Proof for $n=3$:

(

The "figure" for general $n$ is similar, with $n$ right pyramids, one with an $(n-1)$-cube of side length $x_k$ as its base and height $x_k$ for each $k=1,\ldots,n$.)

Sorry about k=1,\ldots,n$. (I made this in Inkscape, a wonderful free-software vector drawing application. For the low qualityinequality and associated labels, but I don't have 3d-ninja-rendering skills. used the textext extension.) 1 [made Community Wiki] Here's a proof of the inequality of the arithmetic and geometric means in the form $$\frac{x_1^n}{n} + \cdots + \frac{x_n^n}{n} \geq x_1\cdots x_n.$$ Proof for$n=3$: (The "figure" for general$n$is similar, with$n$right pyramids, one with an$(n-1)$-cube of side length$x_k$as its base and height$x_k$for each$k=1,\ldots,n\$.)

Sorry about the low quality, but I don't have 3d-ninja-rendering skills.