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  "body": "\n## Statement\n\nLet $I$ be an interval and let $f \\colon I \\to \\mathbb{R}$ be [[convex|0VV0HSW1ZS]].\nLet $x_1, \\dots, x_n \\in I$, and let $w_1, \\dots, w_n \\ge 0$ with $w_1 + \\cdots + w_n = 1$.\nThen\n$$f\\Big( \\sum_{i=1}^{n} w_i x_i \\Big) \\le \\sum_{i=1}^{n} w_i f(x_i).$$\n\n## Remarks\n\nThe case $n = 2$ is the definition of convexity. Induction on $n$ gives the general case.\nFor a concave function, the inequality is reversed. With $f = \\ln$ this gives the [[AM–GM inequality|0DRF0F3XC9]].\nMathlib states it as `ConvexOn.map_sum_le`.\n\n## History\n\nJohan Jensen proved the inequality in 1906.\n\n## Explanations\n\n### L1 human\n\nThe value of a convex function at an average is at most the average of its values.\nThe weights $w_i$ say how much each point counts in the average.\n",
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