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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $K \\subseteq \\R^n$ be a nonempty [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].\nLet $\\operatorname{ext} K$ be the set of [[extreme points|3MSGFJM2NX]] of $K$.\nThen $K$ is the [[convex hull|0FQSDC2EQ2]] of $\\operatorname{ext} K$:\n$$K = \\operatorname{conv}(\\operatorname{ext} K).$$\nMoreover, every point of $K$ is a [[convex combination|21SXYJYC59]] of at most $n + 1$ extreme points of $K$.\n\n## Remarks\n\nProof idea: induct on the dimension of $K$.\nIf $x$ lies on the relative boundary of $K$, the [[supporting hyperplane theorem|DR9BXFWMQB]] gives a hyperplane $H$ through $x$.\nThen $K \\cap H$ is a [[face|KY7SB854VH]] of lower dimension whose extreme points are extreme points of $K$.\nIf $x$ lies in the relative interior, a line through $x$ meets $K$ in a segment with both endpoints on the relative boundary.\nThen $x$ is a convex combination of these endpoints.\nThe bound $n + 1$ then follows from [[Carathéodory's theorem|0201Q4P2B5]].\n\nExample: a filled triangle is the convex hull of its three vertices, and a closed disc is the convex hull of its boundary circle.\n\nBoth hypotheses on $K$ are needed.\nThe closed half-plane $\\set{(x, y) : y \\ge 0}$ is closed and convex but has no extreme points.\nThe open disc is bounded and convex but has no extreme points either.\n\n## Explanations\n\n### L1 human\n\nA compact convex shape, that is, a closed and bounded one, is filled in by its corners.\nHere a corner means an extreme point, a point not in the middle of any segment of the shape.\nFor a polygon the corners are the vertices; for a disc every boundary point counts.\nIn $\\R^n$, every point of the shape is a weighted average of at most $n + 1$ corners, for example at most three in the plane.\n\n### L2 human\n\nThis is the finite-dimensional case of the [[Krein–Milman theorem|RSRAY0Q68S]].\nIn $\\R^n$ no closure is needed: the convex hull of $\\operatorname{ext} K$ is already all of $K$, although $\\operatorname{ext} K$ itself need not be closed when $n \\ge 3$.\nIt explains why a linear function on a compact convex set attains its maximum at an extreme point, which is the basis of the simplex method in linear programming.\nApplied to polytopes, it says that a bounded intersection of finitely many half-spaces is the convex hull of its vertices.\n",
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