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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $C \\subseteq \\R^n$.\nA point $x \\in C$ is an *exposed point* of $C$ if there is a [[linear form|07YQ1C78G0]] $f \\colon \\R^n \\to \\R$ such that $f(x) \u003e f(y)$ for all $y \\in C$ with $y \\ne x$.\n\n## Remarks\n\nEquivalently, there is a [[supporting hyperplane|K7N565NKCV]] $H$ of $C$ with $H \\cap C = \\set{x}$.\nIn a real topological vector space the same definition is used with continuous linear functionals.\n\nEvery exposed point is an [[extreme point|3MSGFJM2NX]].\nSuppose $x = (1 - t) y + t z$ with $y, z \\in C$, $y \\ne z$ and $0 \u003c t \u003c 1$.\nThen $x \\ne y$ and $x \\ne z$, so $f(y) \u003c f(x)$ and $f(z) \u003c f(x)$, and hence $f(x) = (1 - t) f(y) + t f(z) \u003c f(x)$, a contradiction.\n\nExample: in $[0, 1]^2$, the corner $(1, 1)$ is exposed by $f(x_1, x_2) = x_1 + x_2$, since $f \\le 2$ on the square with equality only at $(1, 1)$.\n\nNon-example: let $C$ be the stadium, the convex hull of the discs of radius $1$ centred at $(-1, 0)$ and $(1, 0)$.\nThe point $(1, 1)$ is extreme, but the only supporting line of $C$ through it is $y = 1$, which meets $C$ in the whole segment from $(-1, 1)$ to $(1, 1)$.\nSo no linear form is maximised on $C$ only at $(1, 1)$.\n",
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