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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 $G$ be an [[abelian group|082GHNPJ53]], for example $\\R$.\nA *valuation* on $\\mathcal{K}^n$ is a map $\\varphi \\colon \\mathcal{K}^n \\to G$ such that\n$$\\varphi(K \\cup L) + \\varphi(K \\cap L) = \\varphi(K) + \\varphi(L)$$\nfor all $K, L \\in \\mathcal{K}^n$ with $K \\cup L \\in \\mathcal{K}^n$.\n\n## Remarks\n\nIf $K \\cup L$ is convex, then $K \\cap L$ is nonempty, since a convex set is connected and cannot be the union of two disjoint nonempty closed sets.\nSo $K \\cap L \\in \\mathcal{K}^n$ and the condition makes sense.\nA valuation is *continuous* if it is continuous for the [[Hausdorff distance|F79WYYC07C]], and *invariant under rigid motions* if $\\varphi(gK) = \\varphi(K)$ for every rigid motion $g$ of $\\R^n$.\n\nExample: $\\varphi = \\operatorname{vol}$, Lebesgue measure on $\\R^n$.\nFor measurable sets, $\\operatorname{vol}(K \\cup L) + \\operatorname{vol}(K \\cap L) = \\operatorname{vol}(K) + \\operatorname{vol}(L)$ always holds, so volume is a valuation.\nThe constant function $\\chi(K) = 1$ is a valuation as well; it is the Euler characteristic.\n\nNon-example: $\\varphi(K) = \\operatorname{vol}(K)^2$ on $\\mathcal{K}^1$.\nFor $K = [0, 2]$ and $L = [1, 3]$ one gets $\\varphi(K \\cup L) + \\varphi(K \\cap L) = 9 + 1 = 10$, but $\\varphi(K) + \\varphi(L) = 4 + 4 = 8$.\n\nMany authors include the empty set in $\\mathcal{K}^n$ and require $\\varphi(\\emptyset) = 0$.\nThen the Euler characteristic takes the value $0$ at $\\emptyset$.\n",
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