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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]] and let $B \\subseteq A \\subseteq V$.\nThe set $B$ is an *extreme set* of $A$ if the following holds.\nFor all $x_1, x_2 \\in A$ and $x \\in B$ with $x \\in (x_1, x_2)$, both $x_1$ and $x_2$ lie in $B$.\nHere $(x_1, x_2)$ is the [[open segment|CCZA91NE5X]] between $x_1$ and $x_2$.\n\n## Remarks\n\nA point $x \\in A$ is an [[extreme point|3MSGFJM2NX]] of $A$ exactly when $\\set{x}$ is an extreme set of $A$.\nThe sets $\\emptyset$ and $A$ are always extreme sets of $A$, and an extreme set of an extreme set of $A$ is an extreme set of $A$.\n\nExample: for $A = [0, 1]^2$, the bottom edge $B = [0, 1] \\times \\set{0}$ is an extreme set.\nLet $x \\in B$ lie in $(x_1, x_2)$ with $x_1, x_2 \\in A$.\nThe second coordinates of $x_1, x_2$ are $\\ge 0$, and an average of them with positive weights is $0$.\nSo both are $0$, and $x_1, x_2 \\in B$.\n\nNon-example: the diagonal $D = \\set{(s, s) : 0 \\le s \\le 1}$ of the same square.\nThe point $(1/2, 1/2) \\in D$ lies in the open segment between $(1, 0)$ and $(0, 1)$, which are in $A$ but not in $D$.\n",
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    "name": "Extreme set",
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  "name": "Extreme set",
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