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.
Introduction
Valuations 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 ( φ ( S ) ) = v ( S ) {\displaystyle v(\varphi (S))=v(S)} whenever S ∈ K n {\displaystyle S\in \mathbb {K} ^{n}} and φ {\displaystyle \varphi } is either a translation or a rotation of R n . {\displaystyle \mathbb {R} ^{n}.}
Quermassintegrals
The quermassintegrals W j : K n → R {\displaystyle W_{j}:\mathbb {K} ^{n}\to \mathbb {R} } are defined via Steiner's formula
V 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 0 {\displaystyle W_{0}} 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 Any continuous valuation v {\displaystyle v} on K n {\displaystyle \mathbb {K} ^{n}} that is invariant under rigid motions can be represented as
v ( S ) = ∑ j = 0 n c j W j ( S ) . {\displaystyle v(S)=\sum _{j=0}^{n}c_{j}W_{j}(S)~.}
Corollary 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 Minkowski functional – Function made from a set 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.