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  "body": "\n## Definition\n\nA subset $S$ of $\\mathbb{R}^n$ is convex if for all $x, y \\in S$ and each real number $t$ with $0 \\leq t \\leq 1$,\n$$(1 - t)\\, x + t\\, y \\in S.$$\nThat is, $S$ contains the segment from $x$ to $y$.\n\n## Remarks\n\nThe same definition works in each real vector space or real affine space. Mathlib's `Convex` is stated for a module over an ordered semiring.\nA convex set contains each convex combination $\\sum_i w_i x_i$ of its points, an [[affine combination|0KJH6SRFZK]] with $w_i \\geq 0$.\nThe empty set and the whole space are convex. Each intersection of convex sets is convex, but a union of convex sets need not be convex.\nBalls, half-spaces, lines and triangles with their inside are convex.\n\n## Explanations\n\n### L1 human\n\nA set is convex if you can see each of its points from each other point, with the line of sight inside the set. A disc is convex. A ring is not: the segment between two opposite points of the ring passes through the hole.\n",
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