In
integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the
valuations on
convex bodies in
It was proved by
Hugo Hadwiger
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.
Biography
Although born in Karlsruhe, Ge ...
.
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. A valuation is called invariant under rigid motions if
whenever
and
is either a
translation
Translation is the communication of the semantics, 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 English la ...
or a
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
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 ...
, and
is the constant
is a valuation which is
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
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
Theorems in probability theory