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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $\\mathcal{K}^n$ be the set of [[convex bodies|437FSG3419]] in $\\R^n$, and let $V$ be the volume, that is, [[Lebesgue measure|2Z2QVWZ262]].\n*Brunn–Minkowski theory* is the study of the functionals on $\\mathcal{K}^n$ obtained from $V$ and [[Minkowski addition|SXS6D87PHE]] $K + L = \\set{x + y : x \\in K, \\ y \\in L}$ together with scaling by nonnegative reals.\nIts basis is Minkowski's theorem: for $K_1, \\ldots, K_m \\in \\mathcal{K}^n$ and $\\lambda_1, \\ldots, \\lambda_m \\ge 0$,\n$$V(\\lambda_1 K_1 + \\cdots + \\lambda_m K_m) = \\sum_{i_1, \\ldots, i_n = 1}^{m} \\lambda_{i_1} \\cdots \\lambda_{i_n} V(K_{i_1}, \\ldots, K_{i_n}),$$\nwhere the coefficients $V(K_{i_1}, \\ldots, K_{i_n})$, symmetric in their arguments, are the [[mixed volumes|QB3Q5FBJ08]].\nThe theory studies the mixed volumes and their special cases, and inequalities between them such as the [[Brunn–Minkowski theorem|9B02RPN7XD]].\n\n## Remarks\n\nExample: take $K_1 = K$ and $K_2 = B$, the unit ball.\nThen $V(K + t B)$ is a polynomial of degree $n$ in $t \\ge 0$; this is Steiner's formula.\nIts coefficients are, up to binomial factors, the quermassintegrals of $K$, among them the volume and a multiple of the surface area.\nNon-example: Lutwak's dual Brunn–Minkowski theory replaces Minkowski addition by radial addition of star bodies; it studies sections rather than projections and is a different theory.\nThe $L_p$ Brunn–Minkowski theory is an extension that replaces Minkowski addition by $L_p$ addition of support functions.\n",
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