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Valuation (geometry)

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In geometry, a valuation is a finitely additive function from a collection of subsets of a set X {\displaystyle X} to an abelian semigroup. For example, Lebesgue measure is a valuation on finite unions of convex bodies of R n . {\displaystyle \mathbb {R} ^{n}.} Other examples of valuations on finite unions of convex bodies of R n {\displaystyle \mathbb {R} ^{n}} are surface area, mean width, and Euler characteristic.