About: Valuation (geometry)

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In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic.

Attributes	Values
rdfs:label	Valuation (geometry) (en)
rdfs:comment	In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. (en)
foaf:depiction
dcterms:subject	Geometry
Wikipage page ID	63011965 (xsd:integer)
Wikipage revision ID	1112336438 (xsd:integer)
Link from a Wikipage to another Wikipage	Rolf Schneider Schwartz kernel theorem Beilinson-Bernstein localization Peter McMullen Chern–Gauss–Bonnet theorem General linear group Gauss-Bonnet theorem Geometry Luis Santaló Complex projective space Density on a manifold Embedding Banach space Geometry Wilhelm Blaschke Hausdorff distance Lebesgue measure Locally compact space Semigroup Hilbert's third problem Herbert Federer Mixed volume Differential form Hugo Hadwiger Semyon Alesker Shiing-Shen Chern Euler characteristic Polytope Support function
sameAs	Valuation (geometry) Valuation (geometry)
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has abstract	In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. In the geometric setting, often continuity (or smoothness) conditions are imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory, heavily reliant on advanced tools from abstract algebra. (en)
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foaf:isPrimaryTopicOf	wikipedia-en:Valuation_(geometry)
is rdfs:seeAlso of	Sigma-additive set function
is Link from a Wikipage to another Wikipage of	Valuation (Geometry) Integral geometry Convex geometry Sigma-additive set function Hadwiger's theorem Hilbert's third problem Mixed volume Hugo Hadwiger Semyon Alesker Set function Valuation
is Wikipage redirect of	Valuation (Geometry)
is Wikipage disambiguates of	Valuation
is foaf:primaryTopic of	wikipedia-en:Valuation_(geometry)

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