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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]] and let $x_1, \\dots, x_k$ be points of $V$, with $k \\ge 1$.\nA *convex combination* of $x_1, \\dots, x_k$ is a point of the form\n$$\\lambda_1 x_1 + \\cdots + \\lambda_k x_k,$$\nwhere the real numbers $\\lambda_1, \\dots, \\lambda_k$ satisfy $\\lambda_i \\ge 0$ for each $i$ and $\\lambda_1 + \\cdots + \\lambda_k = 1$.\n\n## Remarks\n\nExample: in $\\R^2$, the point $(1, 1)$ is a convex combination of $(0, 0)$, $(3, 0)$ and $(0, 3)$, with weights $1/3$, $1/3$, $1/3$.\nThe weights are nonnegative and add up to $1$, and $\\tfrac13 (0,0) + \\tfrac13 (3,0) + \\tfrac13 (0,3) = (1, 1)$.\n\nNon-example: $2 \\cdot (1, 0) + (-1) \\cdot (0, 0) = (2, 0)$ uses weights that add up to $1$, so it is an [[affine combination|0KJH6SRFZK]], but the weight $-1$ is negative, so it is not a convex combination.\n\nA convex combination of two points $x, y$ is a point $(1 - t) x + t y$ with $0 \\le t \\le 1$.\nThe same definition works over any ordered field in place of $\\R$.\n",
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