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  "body": "\n## Definition\n\nLet $P$ and $Q$ be [[convex polytopes|300K6CB1KH]] in $\\R^n$.\nTheir *Minkowski sum* is the set\n$$P + Q = \\set{p + q : p \\in P, \\ q \\in Q}.$$\n\n## Remarks\n\nThe set $P + Q$ is again a convex polytope.\nIf $P$ has vertex set $V$ and $Q$ has vertex set $W$, then $P + Q$ is the [[convex hull|0FQSDC2EQ2]] of the finite set $\\set{v + w : v \\in V, \\ w \\in W}$.\nFor a linear functional $u$, the face of $P + Q$ on which $u$ is largest is the sum of the faces of $P$ and of $Q$ on which $u$ is largest.\nSo each vertex of $P + Q$ is $v + w$ for a unique vertex $v$ of $P$ and a unique vertex $w$ of $Q$.\n\nExample: let $T$ be the triangle with vertices $(0,0)$, $(1,0)$, $(0,1)$ in $\\R^2$, and let $-T$ be its image under $x \\mapsto -x$.\nThe sums of vertices of $T$ and $-T$ are the differences of vertices of $T$, namely $(0,0)$, $(\\pm 1, 0)$, $(0, \\pm 1)$, $(1,-1)$ and $(-1,1)$.\nTheir convex hull is the hexagon with vertices $(1,0)$, $(1,-1)$, $(0,-1)$, $(-1,0)$, $(-1,1)$, $(0,1)$, so $T + (-T)$ is this hexagon.\n\nNon-example: the closed unit disk $D$ is convex but not a polytope, and $[0,1]^2 + D$ is a square with rounded corners, which is not a polytope.\nSo the closure of polytopes under the sum needs both summands to be polytopes.\n\nThe operation is commutative and associative, and $\\set{0}$ is its neutral element.\nIt is the general [[Minkowski addition|SXS6D87PHE]] of sets, restricted to polytopes.\n",
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