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

Si vous voulez connaître d'autres articles similaires à Hadwiger's theorem vous pouvez visiter la catégorie Integral geometry.

Laisser un commentaire

Votre adresse email ne sera pas publiée.


Nous utilisons nos propres cookies et ceux de tiers pour améliorer l'expérience utilisateur Plus d'informations