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  "body": "\n## Definition\n\nLet $P \\subset \\R^d$ be a [[convex polytope|300K6CB1KH]].\nA *vertex* of $P$ is a [[face|KY7SB854VH]] of $P$ of dimension $0$, identified with the single point it contains.\nEquivalently, a point $v \\in P$ is a vertex if there is a vector $c \\in \\R^d$ such that $c \\cdot v \u003e c \\cdot x$ for every $x \\in P$ with $x \\ne v$.\n\n## Remarks\n\nExample: in the square $P = [0,1]^2$, the point $v = (1,1)$ is a vertex.\nTake $c = (1,1)$: then $c \\cdot v = 2$, and $c \\cdot x = x_1 + x_2 \u003c 2$ for every other point $x$ of $P$.\n\nNon-example: the point $(1, \\tfrac12)$ of the same square is not a vertex.\nFor any $c$, the points $(1, \\tfrac12 \\pm t)$ for small $t \u003e 0$ lie in $P$, and $c \\cdot x$ is at least $c \\cdot (1,\\tfrac12)$ at one of them, so no $c$ singles it out.\n\nThe vertices of $P$ are exactly its [[extreme points|3MSGFJM2NX]].\nIf $V$ is the set of vertices, then $P$ is the [[convex hull|0FQSDC2EQ2]] of $V$, and every finite set whose convex hull is $P$ contains $V$.\n",
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