In
integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformation ...
(otherwise called geometric probability theory), Hadwiger's theorem characterises the
valuations on
convex bodies
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior.
A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...
in
It was proved by
Hugo Hadwiger
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.
Biography
Although born in Karlsruhe, Germany, Hadwige ...
.
Introduction
Valuations
Let
be the collection of all compact convex sets in
A valuation is a function
such that
and for every
that satisfy
A valuation is called continuous if it is continuous with respect to the
Hausdorff metric In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metri ...
. A valuation is called invariant under rigid motions if
whenever
and
is either a
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
or a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
of
Quermassintegrals
The quermassintegrals
are defined via Steiner's formula
where
is the Euclidean ball. For example,
is the volume,
is proportional to the
surface measure,
is proportional to the
mean width In geometry, the mean width is a measure of the "size" of a body; see Hadwiger's theorem for more about the available measures of bodies. In n dimensions, one has to consider (n-1)-dimensional hyperplanes perpendicular to a given direction \hat in S ...
, and
is the constant
is a valuation which is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
of degree
that is,
Statement
Any continuous valuation
on
that is invariant under rigid motions can be represented as
Corollary
Any continuous valuation
on
that is invariant under rigid motions and homogeneous of degree
is a multiple of
See also
*
*
References
An account and a proof of Hadwiger's theorem may be found in
*
An elementary and self-contained proof was given by Beifang Chen in
* {{cite journal, title=A simplified elementary proof of Hadwiger's volume theorem, journal=Geom. Dedicata, volume=105, year=2004, pages=107–120, last=Chen, first=B., mr=2057247, doi=10.1023/b:geom.0000024665.02286.46
Integral geometry
Theorems in convex geometry
Probability theorems