Hadwiger's theorem

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

Inhalt 1 Einführung 1.1 Valuations 1.2 Quermassintegrals 2 Aussage 2.1 Logische Folge 3 Siehe auch 4 References Introduction Valuations Let {Anzeigestil mathbb {K} ^{n}} be the collection of all compact convex sets in {Anzeigestil mathbb {R} ^{n}.} A valuation is a function {Anzeigestil v:mathbb {K} ^{n}zu mathbb {R} } so dass {Anzeigestil v(varnothing )=0} and for every {Anzeigestil S,Tin mathbb {K} ^{n}} that satisfy {displaystyle Scup Tin mathbb {K} ^{n},} {Anzeigestil v(S)+v(T)=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 {Anzeigestil v(Varphi (S))=v(S)} wann immer {displaystyle Sin mathbb {K} ^{n}} und {Anzeigestil Varphi } is either a translation or a rotation of {Anzeigestil mathbb {R} ^{n}.} Quermassintegrals Main article: quermassintegral The quermassintegrals {Anzeigestil W_{j}:mathbb {K} ^{n}zu mathbb {R} } are defined via Steiner's formula {Anzeigestil mathrm {Vol} _{n}(K+tB)= Summe _{j=0}^{n}{manche von ihnen {n}{j}}W_{j}(K)t^{j}~,} wo {Anzeigestil B} is the Euclidean ball. Zum Beispiel, {Anzeigestil W_{Ö}} is the volume, {Anzeigestil W_{1}} is proportional to the surface measure, {Anzeigestil W_{n-1}} is proportional to the mean width, und {Anzeigestil W_{n}} is the constant {Anzeigestil Betreibername {Vol} _{n}(B).} {Anzeigestil W_{j}} is a valuation which is homogeneous of degree {displaystyle n-j,} das ist, {Anzeigestil W_{j}(tK)=t^{n-j}W_{j}(K)~,quad tgeq 0~.} Statement Any continuous valuation {Anzeigestil v} an {Anzeigestil mathbb {K} ^{n}} that is invariant under rigid motions can be represented as {Anzeigestil v(S)= Summe _{j=0}^{n}c_{j}W_{j}(S)~.} Corollary Any continuous valuation {Anzeigestil v} an {Anzeigestil mathbb {K} ^{n}} that is invariant under rigid motions and homogeneous of degree {Anzeigestil j} is a multiple of {Anzeigestil 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.; Rota, G.-C. (1997). Introduction to geometric probability. Cambridge: Cambridge University Press. ISBN 0-521-59362-X. HERR 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. HERR 2057247. Kategorien: Integral geometryTheorems in convex geometryProbability theorems