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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $S \\subseteq \\R^n$.\nA hyperplane $H = \\set{x \\in \\R^n : a \\cdot x = b}$, with $a \\in \\R^n \\setminus \\set{0}$ and $b \\in \\R$, is a *supporting hyperplane* of $S$ if:\n\n1. $S$ lies in one of the closed half-spaces of $H$, say $a \\cdot x \\le b$ for all $x \\in S$;\n2. $H$ contains at least one point of the [[boundary|05D5EGMFYV]] of $S$.\n\nHere $a \\cdot x$ is the [[dot product|0S52TNNTH9]].\nIf $x_0 \\in H$ is such a boundary point, $H$ supports $S$ at $x_0$.\n\n## Remarks\n\nExample: for the closed unit disc $D \\subseteq \\R^2$ and $a = (0, 1)$, $b = 1$, the line $y = 1$ supports $D$ at $(0, 1)$.\nEach $(x, y) \\in D$ has $y \\le 1$, and $(0, 1)$ is a boundary point of $D$ on the line.\n\nNon-example: the line $y = 2$ satisfies clause 1 for $D$, but it misses $D$ and its boundary, so clause 2 fails.\nThe line $y = 0$ meets the boundary of $D$ but has points of $D$ on both sides, so clause 1 fails.\n\nIn terms of the [[support function|1H92CYA4ZC]] $h_S$, when $S$ is closed and bounded and nonempty, the hyperplane $a \\cdot x = h_S(a)$ is the supporting hyperplane with outer normal $a$.\n",
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