Let be the collection of all compact convex sets in A valuation is a function such that and for every that satisfy
A valuation is called continuous if it is continuous with respect to the Hausdorff metric. A valuation is called invariant under rigid motions if whenever and is either a translation or a rotation of
The quermassintegrals are defined via Steiner's formula
where is the Euclidean ball. For example, is the volume, is proportional to the surface measure, is proportional to the mean width, and is the constant
is a valuation which is homogeneous of degree that is,
Statement
Any continuous valuation on that is invariant under rigid motions can be represented as
Corollary
Any continuous valuation on that is invariant under rigid motions and homogeneous of degree is a multiple of
An account and a proof of Hadwiger's theorem may be found in
Klain, D.A.; Rota, G.-C. (1997). Introduction to geometric probability. Cambridge: Cambridge University Press. ISBN0-521-59362-X. MR 1608265.
An elementary and self-contained proof was given by Beifang Chen in
Chen, B. (2004). "A simplified elementary proof of Hadwiger's volume theorem". Geom. Dedicata. 105: 107–120. doi:10.1023/b:geom.0000024665.02286.46. MR 2057247.
January 11, 2023
hadwiger, theorem, integral, geometry, otherwise, called, geometric, probability, theory, characterises, valuations, convex, bodies, displaystyle, mathbb, proved, hugo, hadwiger, contents, introduction, valuations, quermassintegrals, statement, corollary, also. In integral geometry otherwise called geometric probability theory Hadwiger s theorem characterises the valuations on convex bodies in R n displaystyle mathbb R n It was proved by Hugo Hadwiger Contents 1 Introduction 1 1 Valuations 1 2 Quermassintegrals 2 Statement 2 1 Corollary 3 See also 4 ReferencesIntroduction EditValuations Edit Let K n displaystyle mathbb K n be the collection of all compact convex sets in R n displaystyle mathbb R n A valuation is a function v K n R displaystyle v mathbb K n to mathbb R such that v 0 displaystyle v varnothing 0 and for every S T K n displaystyle S T in mathbb K n that satisfy S T K n displaystyle S cup T in mathbb K n v S v T v S T v S T displaystyle v S v T v S cap T v S cup T A valuation is called continuous if it is continuous with respect to the Hausdorff metric A valuation is called invariant under rigid motions if v f S v S displaystyle v varphi S v S whenever S K n displaystyle S in mathbb K n and f displaystyle varphi is either a translation or a rotation of R n displaystyle mathbb R n Quermassintegrals Edit Main article quermassintegral The quermassintegrals W j K n R displaystyle W j mathbb K n to mathbb R are defined via Steiner s formulaV o l n K t B j 0 n n j W j K t j displaystyle mathrm Vol n K tB sum j 0 n binom n j W j K t j where B displaystyle B is the Euclidean ball For example W o displaystyle W o is the volume W 1 displaystyle W 1 is proportional to the surface measure W n 1 displaystyle W n 1 is proportional to the mean width and W n displaystyle W n is the constant Vol n B displaystyle operatorname Vol n B W j displaystyle W j is a valuation which is homogeneous of degree n j displaystyle n j that is W j t K t n j W j K t 0 displaystyle W j tK t n j W j K quad t geq 0 Statement EditAny continuous valuation v displaystyle v on K n displaystyle mathbb K n that is invariant under rigid motions can be represented asv S j 0 n c j W j S displaystyle v S sum j 0 n c j W j S Corollary Edit Any continuous valuation v displaystyle v on K n displaystyle mathbb K n that is invariant under rigid motions and homogeneous of degree j displaystyle j is a multiple of W n j displaystyle W n j See also EditMinkowski functional Set function Function from sets to numbersReferences EditAn account and a proof of Hadwiger s theorem may be found in Klain D A Rota G C 1997 Introduction to geometric probability Cambridge Cambridge University Press ISBN 0 521 59362 X MR 1608265 An elementary and self contained proof was given by Beifang Chen in Chen B 2004 A simplified elementary proof of Hadwiger s volume theorem Geom Dedicata 105 107 120 doi 10 1023 b geom 0000024665 02286 46 MR 2057247 Retrieved from https en wikipedia org w index php title Hadwiger 27s theorem amp oldid 1053420154, wikipedia, wiki, book, books, library,