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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]] and let $A, B \\subseteq V$.\nThe *Minkowski sum* of $A$ and $B$ is the set\n$$A + B = \\set{a + b : a \\in A, \\ b \\in B}.$$\nFor a real number $\\lambda$, the *dilate* of $A$ is $\\lambda A = \\set{\\lambda a : a \\in A}$.\nThe operation $(A, B) \\mapsto A + B$ is *Minkowski addition*.\n\n## Remarks\n\nExample: in $\\R^2$, let $A = [0,1] \\times \\set{0}$ and $B = \\set{0} \\times [0, 1]$.\nA point $(s, 0) + (0, t)$ with $s, t \\in [0, 1]$ is $(s, t)$, so $A + B$ is the unit square $[0,1]^2$.\n\nMinkowski addition is associative and commutative, with neutral element $\\set{0}$, and $A + \\emptyset = \\emptyset$.\nIf $A$ and $B$ are convex, then $A + B$ is convex; if they are compact, then $A + B$ is compact.\n\nNon-example of a familiar rule: $A + A = 2A$ can fail.\nFor $A = \\set{0, 1} \\subseteq \\R$ one has $A + A = \\set{0, 1, 2}$ but $2A = \\set{0, 2}$.\nFor convex $A$ and $\\lambda, \\mu \\ge 0$ one has $\\lambda A + \\mu A = (\\lambda + \\mu) A$.\nIn terms of [[support functions|1H92CYA4ZC]], $h_{A + B} = h_A + h_B$ for nonempty compact convex $A, B \\subseteq \\R^n$.\nFor polytopes, see the [[Minkowski sum of polytopes|7GK5417BJF]].\n",
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