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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $K \\subseteq \\R^n$ be a [[compact|0R6EWFKRCF]] [[convex set|02AS45T3H9]].\nThen every [[extreme point|3MSGFJM2NX]] of $K$ is a limit of [[exposed points|VN4M72SQW5]] of $K$.\nThat is, the set of extreme points of $K$ lies in the [[closure|0QEAJB91GZ]] of the set of exposed points of $K$.\n\n## Remarks\n\nProof idea: for a point $z \\in \\R^n$, a point $p$ of $K$ farthest from $z$ is exposed.\nIndeed, the sphere around $z$ through $p$ encloses $K$, and its tangent hyperplane at $p$ meets $K$ only in $p$.\nOne then shows that every extreme point can be approached by such farthest points, by choosing $z$ suitably far away in suitable directions.\n\nCombined with the [[Minkowski–Carathéodory theorem|NT4SKSYKG9]], it follows that $K$ is the [[convex hull|0FQSDC2EQ2]] of the closure of its exposed points.\n\nExample: let $K$ be the stadium, the convex hull of the discs of radius $1$ centred at $(-1, 0)$ and $(1, 0)$.\nThe four points $(\\pm 1, \\pm 1)$ are extreme but not exposed.\nEach of them is a limit of points on the round ends, and those points are exposed.\n\n## Explanations\n\n### L1 human\n\nExtreme points are corners in a broad sense, and exposed points are corners that a flat wall can touch alone.\nSome corners are not exposed, such as the points where the straight sides of a stadium meet its round ends.\nThe theorem says these exceptions are rare: each extreme point is a limit of exposed points.\n\n### L2 human\n\nThe theorem lets one replace extreme points by exposed points in many arguments, at the cost of a closure.\nExposed points are often easier to handle, since each comes with a linear function maximised exactly there.\nIt is used in the study of faces of convex bodies and in convex analysis, where exposed points correspond to directions in which the support function is differentiable.\n",
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