Hadwiger's theorem

Hadwiger's theorem In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in {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 References Introduction Valuations Let {displaystyle mathbb {K} ^{n}} be the collection of all compact convex sets in {displaystyle mathbb {R} ^{n}.} A valuation is a function {displaystyle v:mathbb {K} ^{n}to mathbb {R} } such that {displaystyle v(varnothing )=0} and for every {displaystyle S,Tin mathbb {K} ^{n}} that satisfy {displaystyle Scup Tin mathbb {K} ^{n},} {displaystyle 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 {displaystyle v(varphi (S))=v(S)} whenever {displaystyle Sin mathbb {K} ^{n}} and {displaystyle varphi } is either a translation or a rotation of {displaystyle mathbb {R} ^{n}.} Quermassintegrals Main article: quermassintegral The quermassintegrals {displaystyle W_{j}:mathbb {K} ^{n}to mathbb {R} } are defined via Steiner's formula {displaystyle mathrm {Vol} _{n}(K+tB)=sum _{j=0}^{n}{binom {n}{j}}W_{j}(K)t^{j}~,} where {displaystyle B} is the Euclidean ball. For example, {displaystyle W_{o}} is the volume, {displaystyle W_{1}} is proportional to the surface measure, {displaystyle W_{n-1}} is proportional to the mean width, and {displaystyle W_{n}} is the constant {displaystyle operatorname {Vol} _{n}(B).} {displaystyle W_{j}} is a valuation which is homogeneous of degree {displaystyle n-j,} that is, {displaystyle W_{j}(tK)=t^{n-j}W_{j}(K)~,quad tgeq 0~.} Statement Any continuous valuation {displaystyle v} on {displaystyle mathbb {K} ^{n}} that is invariant under rigid motions can be represented as {displaystyle v(S)=sum _{j=0}^{n}c_{j}W_{j}(S)~.} Corollary Any continuous valuation {displaystyle v} on {displaystyle mathbb {K} ^{n}} that is invariant under rigid motions and homogeneous of degree {displaystyle j} is a multiple of {displaystyle 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. 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. Categories: Integral geometryTheorems in convex geometryProbability theorems