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  "body": "\n## Definition\n\nLet $X$ be a [[vector space|000AYPA9AJ]] over $\\R$ or $\\C$, and let $K \\subseteq X$ be a [[convex set|02AS45T3H9]].\nA point $x \\in K$ is an *extreme point* of $K$ if, whenever $x = t y + (1 - t) z$ with $y, z \\in K$ and $0 \u003c t \u003c 1$, one has $y = z = x$.\nThe set of extreme points of $K$ is written $\\operatorname{ext} K$.\n\n## Remarks\n\nIn $\\R^2$, let $K = [0, 1]^2$.\nThe corner $(0, 0)$ is extreme: if $(0, 0) = t y + (1 - t) z$ with $y, z \\in K$ and $0 \u003c t \u003c 1$, then each coordinate is a combination of nonnegative numbers with positive weights equal to $0$, so $y = z = (0, 0)$.\nThe point $(1/2, 0)$ is not extreme, because it is the midpoint of $(0, 0)$ and $(1, 0)$, which lie in $K$ and differ from it.\nEquivalently, $x$ is extreme when $x$ is not the midpoint of two distinct points of $K$.\nAn open ball in a nonzero normed space has no extreme points.\nThe closed unit ball of the sequence space $c_0$ also has no extreme points.\n",
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