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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $\\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$.\nLet $A, B \\subseteq \\R^n$ be nonempty [[compact sets|0R6EWFKRCF]], and let $A + B = \\set{a + b : a \\in A, \\ b \\in B}$ be their Minkowski sum.\nThen\n$$\\lambda(A + B)^{1/n} \\ge \\lambda(A)^{1/n} + \\lambda(B)^{1/n}.$$\n\n## Remarks\n\nFor compact $A$ and $B$ the set $A + B$ is compact, hence measurable; for general measurable sets $A + B$ need not be measurable.\nOne proof first checks boxes, where the inequality is the AM–GM inequality for the side ratios.\nIt then passes to finite unions of disjoint boxes by induction on the number of boxes, cutting both sets by a coordinate hyperplane in matching proportions.\nCompact sets are then approximated from outside by such unions.\nAnother proof applies the [[Prékopa–Leindler inequality|J9CJX9F42Y]] to indicator functions.\nFor compact convex sets of positive volume, equality holds if and only if $A$ and $B$ are homothetic.\nExample: for balls of radii $r$ and $s$, $A + B$ is a ball of radius $r + s$, and both sides equal $\\omega_n^{1/n} (r + s)$, where $\\omega_n$ is the volume of the unit ball.\n\n## Explanations\n\n### L1 human\n\nAdd two shapes by adding every point of one to every point of the other.\nThe theorem says that the volume of the sum is at least as large as a natural guess.\nAfter taking $n$-th roots, volume behaves at least additively under this addition.\nThis one inequality hides many facts about volume and shape.\n\n### L2 human\n\nThe function $t \\mapsto \\lambda((1-t) A + t B)^{1/n}$ is concave on $[0, 1]$ for convex $A$ and $B$; this concavity form goes back to Brunn.\nApplied to $A$ and a small ball, the theorem gives the [[isoperimetric inequality|KBX8RBZ626]].\nIt is the starting point of the theory of mixed volumes, and it has a functional form, the Prékopa–Leindler inequality, used in probability and convex geometry.\n",
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    "name": "Brunn–Minkowski theorem",
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