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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\mathcal{K}^n$ be the set of nonempty compact convex subsets of $\\R^n$ (the [[convex bodies|437FSG3419]] in the broad sense).\nLet $\\operatorname{vol}$ be [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$.\nThere is a unique function $V \\colon (\\mathcal{K}^n)^n \\to \\R$ such that:\n\n1. $V$ is symmetric in its $n$ arguments;\n2. for every $r \\ge 1$, all $K_1, \\dots, K_r \\in \\mathcal{K}^n$ and all $\\lambda_1, \\dots, \\lambda_r \\ge 0$,\n$$\\operatorname{vol}(\\lambda_1 K_1 + \\cdots + \\lambda_r K_r) = \\sum_{i_1 = 1}^{r} \\cdots \\sum_{i_n = 1}^{r} V(K_{i_1}, \\dots, K_{i_n}) \\, \\lambda_{i_1} \\cdots \\lambda_{i_n},$$\nwhere the sum on the left is the [[Minkowski sum|SXS6D87PHE]].\n\nThe number $V(K_1, \\dots, K_n)$ is the *mixed volume* of $K_1, \\dots, K_n$.\n\n## Remarks\n\nThe left side of clause 2 is a [[homogeneous polynomial|2C3PAQ7H50]] of degree $n$ in $\\lambda_1, \\dots, \\lambda_r$; this is the theorem of Minkowski that makes the definition possible.\nAn explicit formula is\n$$V(K_1, \\dots, K_n) = \\frac{1}{n!} \\sum_{\\emptyset \\ne J \\subseteq \\set{1, \\dots, n}} (-1)^{n - \\abs{J}} \\operatorname{vol}\\Big(\\sum_{j \\in J} K_j\\Big).$$\nMixed volumes are nonnegative, invariant under translating each argument, linear in each argument for Minkowski addition with nonnegative scalars, and monotone under inclusion.\nAlso $V(K, \\dots, K) = \\operatorname{vol}(K)$.\n\nExample: $n = 2$, $K$ a compact convex set of area $A$ and perimeter $L$, and $D$ the closed unit disk.\nThe area of $K + tD$ is $A + L t + \\pi t^2$, while clause 2 gives $A + 2 V(K, D)\\, t + \\pi t^2$.\nSo $V(K, D) = L / 2$.\n\nNon-example: without clause 1 the coefficients are not unique.\nFor $n = 2$ one may move all of the coefficient $2 V(K_1, K_2)$ to the term $(i_1, i_2) = (1, 2)$ and put $0$ at $(2, 1)$, which still satisfies clause 2.\n",
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