Hadwiger's theorem
Encyclopedia
In integral geometry
(otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations
on convex bodies
in Rn. It was proved by Hugo Hadwiger
.
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) whenever S ∈ Kn and φ is either a translation
or a rotation
of Rn.
where B is the Euclidean ball. For example, W0 is the volume, W1 is proportional to the surface measure
, Wn-1 is proportional to the mean width
, and Wn is the constant Voln(B).
Wj is a valuation which is homogeneous
of degree n-j, that is,
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 transformations from the space of functions on one geometrical space to the...
(otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations
Valuation (measure theory)
In measure theory or at least in the approach to it through domain theory, a valuation is a map from the class of open sets of a topological space to the set positive real numbers including infinity...
on convex bodies
Convex body
In mathematics, a convex body in n-dimensional Euclidean space Rn 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, i.e. a point x lies in K if and only if its antipode, −x, also lies in K...
in Rn. It was proved by Hugo Hadwiger
Hugo Hadwiger
Hugo Hadwiger was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.-Biography:...
.
Valuations
Let Kn be the collection of all convex bodies in Rn. A valuation is a function v:Kn → R such that v(∅) = 0 and, for every S,T ∈Kn for which S∪T∈Kn,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) whenever S ∈ Kn and φ is either a translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
or a rotation
Rotation (mathematics)
In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
of Rn.
Quermassintegrals
The quermassintegrals Wj: Kn → R are defined via Steiner's formulawhere B is the Euclidean ball. For example, W0 is the volume, W1 is proportional to the surface measure
Minkowski content
The Minkowski content of a set, or the boundary measure, is a basic concept in geometry and measure theory which generalizes to arbitrary measurable sets the notions of length of a smooth curve in the plane and area of a smooth surface in the space...
, Wn-1 is proportional to the mean width
Mean width
In geometry, the mean width is a measure of dimension length of the "size" a body; see Hadwiger's theorem for more about the available measures of bodies...
, and Wn is the constant Voln(B).
Wj is a valuation which is homogeneous
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
of degree n-j, that is,
Statement
Any continuous valuation v on Kn that is invariant under rigid motions can be represented as