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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\mathcal{K}^n$ be the set of nonempty compact convex subsets of $\\R^n$, with the [[Hausdorff distance|F79WYYC07C]].\nLet $\\varphi \\colon \\mathcal{K}^n \\to \\R$ be a [[valuation|V0RBKHYTV7]] that is continuous and invariant under the [[rigid motions|0H3YY8CPKH]] of $\\R^n$.\nThen there are real numbers $c_0, c_1, \\dots, c_n$ such that\n$$\\varphi(K) = \\sum_{j=0}^n c_j V_j(K) \\quad \\text{for all } K \\in \\mathcal{K}^n,$$\nwhere $V_0, \\dots, V_n$ are the [[intrinsic volumes|C2CK6KSSTV]].\n\n## Remarks\n\nConversely, each $V_j$ is a continuous valuation invariant under rigid motions, so these valuations form a real vector space of dimension $n + 1$ with basis $V_0, \\dots, V_n$.\nA short proof by Klain rests on a lemma about simple valuations.\nThe lemma says: a continuous valuation invariant under rigid motions that vanishes on all sets of dimension below $n$ is a multiple of volume.\nInduction on $n$ then handles the restriction of $\\varphi$ to lower-dimensional sets.\n\nExample: surface area in $\\R^n$ is a continuous valuation invariant under rigid motions, and the theorem identifies it as $2 V_{n-1}$.\n\nInvariance under rotations is needed.\nIn $\\R^2$ the width in the direction of the first axis, $\\varphi(K) = \\max_{x \\in K} x_1 - \\min_{x \\in K} x_1$, is a continuous valuation invariant under translations.\nIt gives $1$ to the segment $[0,1] \\times \\set{0}$ and $0$ to the congruent segment $\\set{0} \\times [0, 1]$, so it is not a combination of intrinsic volumes.\n\n## Explanations\n\n### L1 human\n\nThere are many ways to measure the size of a convex shape.\nSuppose a measure is additive like an area, does not change when the shape is moved, and changes little when the shape changes little.\nHadwiger's theorem says that such a measure is built from a short list of basic measures.\nIn space the list has four entries: volume, surface area, mean width and the Euler characteristic.\n\n### L2 human\n\nThe theorem is the foundation of integral geometry in Euclidean space.\nIt gives quick proofs of the principal kinematic formula and of Crofton-type formulas: one checks that both sides are such valuations and compares them on balls.\nIt also explains why intrinsic volumes appear in probability, for example in the expected Euler characteristic of random sets.\nIts classification has been extended by Alesker and others to valuations invariant under smaller groups.\n",
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