Hadwiger's theorem (integral geometry)

The set of Hadwiger is a theorem from the mathematical field of integral geometry . It says that every continuous and under isometry invariant evaluation of compact, convex subsets of the is a linear combination of transversal integrals. ${\displaystyle \mathbb {R} ^{n}}$

Terms

A continuous evaluation is a real-valued functional on the set of all compact , convex subsets with and for all , which is continuous with respect to the Hausdorff metric . ${\displaystyle v}$ ${\displaystyle K\subset \mathbb {R} ^{n}}$ ${\displaystyle v(\emptyset )=0}$ ${\displaystyle v(S\cup T)+v(S\cap T)=v(S)+v(T)}$ ${\displaystyle S,T}$

The transversal integrals are functionals that act as coefficients of the power series expansion ${\displaystyle W_{0},\ldots ,W_{n}}$

${\displaystyle \mathrm {Vol} _{n}(K+tB^{n})=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}}$

are defined for the unit sphere and every compact, convex body . ${\displaystyle B^{n}}$ ${\displaystyle K\subset \mathbb {R} ^{n}}$

Hadwiger's theorem

Every continuous assessment , which is invariant under all isometries of is, is a linear combination of Quermaßintegralen: ${\displaystyle v}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle v(K)=\sum _{j=0}^{n}c_{j}W_{j}(K)}$

with independent coefficients . ${\displaystyle K}$ ${\displaystyle c_{j}}$

literature

• DA Klain, G.-C. Rota: Introduction to Geometric Probability . Cambridge University Press, Cambridge 1997, ISBN 0-521-59362-X .
• B. Chen: A simplified elementary proof of Hadwiger's volume theorem . In: Geom. Dedicata . 105, 2004, pp. 107-120. doi : 10.1023 / b: geom.0000024665.02286.46 .