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  "body": "\n## Statement\n\nLet $X$ be a Hausdorff [[locally convex space|B6GY2KJMWW]] over $\\R$ or $\\C$, and let $K \\subseteq X$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].\nThen $K$ has at least one [[extreme point|3MSGFJM2NX]].\nMoreover, $K$ is the [[closure|0QEAJB91GZ]] of the [[convex hull|0FQSDC2EQ2]] of the set $\\operatorname{ext} K$ of its extreme points:\n$$K = \\overline{\\operatorname{conv}}(\\operatorname{ext} K).$$\n\n## Remarks\n\nCall a nonempty closed subset $F \\subseteq K$ a face if every open segment of $K$ that meets $F$ has both endpoints in $F$.\nBy [[Zorn's lemma|04RXTQA5ZA]] and compactness, every face contains a minimal face.\nA minimal face is a single point, since otherwise a continuous functional that separates two of its points would cut out a smaller face; that point is extreme.\nIf $\\overline{\\operatorname{conv}}(\\operatorname{ext} K)$ were smaller than $K$, the [[Hahn–Banach separation theorem|6GWMGFGFJJ]] would give a continuous functional whose real part has its maximum on $K$ only on a face disjoint from $\\operatorname{ext} K$.\nBut that face contains a minimal face, which is an extreme point of $K$, a contradiction.\n\nFor the square $[0, 1]^2$ in $\\R^2$ the theorem says that the square is the convex hull of its four corners.\nA standard application: the closed unit ball of a dual Banach space is weak*-compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], so it has extreme points.\nThe unit ball of $c_0$ has no extreme points, so $c_0$ is not isometrically isomorphic to the dual of any Banach space.\nCompactness is needed: the closed half-plane $\\set{(x, y) : y \\ge 0}$ in $\\R^2$ is closed and convex but has no extreme points.\n\n## Explanations\n\n### L1 human\n\nA compact convex set is determined by its corners.\nTaking all mixtures of the corners, and then all limits of such mixtures, gives back the whole set.\nFor a polygon this is clear; the theorem says the same in infinite-dimensional spaces.\n\n### L2 human\n\nThe theorem is used to prove that certain Banach spaces are not dual spaces, by showing that their unit balls lack extreme points.\nIn measure theory and probability, the extreme points of compact convex sets of measures or states are often the \"pure\" or \"ergodic\" objects.\nThe theorem shows that they generate everything.\nChoquet theory refines the closed convex hull into an integral representation over the extreme points.\nThe proof combines Zorn's lemma with the [[Hahn–Banach separation theorem|6GWMGFGFJJ]].\n",
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