Hadwiger's theorem

Hadwiger's theorem In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in {style d'affichage mathbb {R} ^{n}.} It was proved by Hugo Hadwiger.

Contenu 1 Introduction 1.1 Valuations 1.2 Quermassintegrals 2 Déclaration 2.1 Corollaire 3 Voir également 4 References Introduction Valuations Let {style d'affichage mathbb {K} ^{n}} be the collection of all compact convex sets in {style d'affichage mathbb {R} ^{n}.} A valuation is a function {style d'affichage v:mathbb {K} ^{n}à mathbb {R} } tel que {style d'affichage v(varnothing )=0} and for every {style d'affichage S,Tin mathbb {K} ^{n}} that satisfy {displaystyle Scup Tin mathbb {K} ^{n},} {style d'affichage v(S)+v(J)=v(Scap T)+v(Scup 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 {style d'affichage v(varphi (S))=v(S)} chaque fois que {displaystyle Sin mathbb {K} ^{n}} et {style d'affichage varphi } is either a translation or a rotation of {style d'affichage mathbb {R} ^{n}.} Quermassintegrals Main article: quermassintegral The quermassintegrals {style d'affichage W_{j}:mathbb {K} ^{n}à mathbb {R} } are defined via Steiner's formula {style d'affichage mathrm {Volume} _{n}(K+tB)=somme _{j=0}^{n}{certains d'entre eux {n}{j}}W_{j}(K)t ^{j}~,} où {style d'affichage B} is the Euclidean ball. Par exemple, {style d'affichage W_{o}} is the volume, {style d'affichage W_{1}} is proportional to the surface measure, {style d'affichage W_{n-1}} is proportional to the mean width, et {style d'affichage W_{n}} is the constant {nom de l'opérateur de style d'affichage {Volume} _{n}(B).} {style d'affichage W_{j}} is a valuation which is homogeneous of degree {displaystyle n-j,} C'est, {style d'affichage W_{j}(tK)=t^{n-j}W_{j}(K)~,quad tgeq 0~.} Statement Any continuous valuation {style d'affichage v} sur {style d'affichage mathbb {K} ^{n}} that is invariant under rigid motions can be represented as {style d'affichage v(S)=somme _{j=0}^{n}c_{j}W_{j}(S)~.} Corollary Any continuous valuation {style d'affichage v} sur {style d'affichage mathbb {K} ^{n}} that is invariant under rigid motions and homogeneous of degree {displaystyle j} is a multiple of {style d'affichage W_{n-j}.} See also Minkowski functional Set function – Function from sets to numbers References An account and a proof of Hadwiger's theorem may be found in Klain, D.A.; Rotation, G.-C. (1997). Introduction to geometric probability. Cambridge: la presse de l'Universite de Cambridge. ISBN 0-521-59362-X. M 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". Géom. Dedicata. 105: 107–120. est ce que je:10.1023/b:geom.0000024665.02286.46. M 2057247. Catégories: Integral geometryTheorems in convex geometryProbability theorems