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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $x_1, \\dots, x_n$ be nonnegative real numbers.\nThen\n$$\\sqrt[n]{x_1 x_2 \\cdots x_n} \\le \\frac{x_1 + x_2 + \\dots + x_n}{n},$$\nwith equality if and only if $x_1 = x_2 = \\dots = x_n$.\n\n## Remarks\n\nFor $n = 2$, the inequality $\\sqrt{x y} \\le (x + y)/2$ follows from $(\\sqrt{x} - \\sqrt{y})^2 \\ge 0$.\nThe weighted form: for weights $w_i \\ge 0$ with $w_1 + \\dots + w_n = 1$, $x_1^{w_1} \\cdots x_n^{w_n} \\le w_1 x_1 + \\dots + w_n x_n$.\n\n## Explanations\n\n### L1 human\n\nThe geometric mean of some nonnegative numbers is never larger than their average.\nFor example, for $2$ and $8$, the geometric mean is $\\sqrt{16} = 4$ and the average is $5$.\nAmong all rectangles with a fixed perimeter, the square has the largest area. This is the case $n = 2$.\n",
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