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  "body": "\n## Summary\n\nA midpoint convex function, also called a Jensen-convex function, is a real function $f$ on an interval with $f\\big(\\tfrac{x + y}{2}\\big) \\le \\tfrac{f(x) + f(y)}{2}$ for all $x, y$.\nEvery [[convex function|07Q7CNM5WF]] is midpoint convex, but the converse fails without some regularity.\nMild extra conditions close the gap: a midpoint convex function that is bounded above on a small interval, or that is measurable, is convex.\nThe notion is the starting point of the theory of convex functions as solutions of a [[functional inequality|YQB2B0EXY8]].\n\n## Explanations\n\n### L1 human\n\nConvexity asks that every chord of the graph lies above the graph.\nMidpoint convexity only asks this at the midpoint of each chord.\nFor nice functions this is the same thing, because midpoints of midpoints fill out the whole chord.\nStrange functions built with the axiom of choice show that, in general, the midpoint condition alone is weaker.\n",
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