In geometry, a valuation is a finitely additive function from a collection of subsets of a set
For example, Lebesgue measure is a valuation on finite unions of convex bodies of
Other examples of valuations on finite unions of convex bodies of
In geometry, continuity (or smoothness) conditions are often imposed on valuations, but there are also purely discrete facets of the theory.
In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory reliant on tools from abstract algebra.
Valuations are not only graded by the degree of homogeneity, but also by the parity with respect to the reflection through the origin, namely
-homogeneous valuations, into the space of continuous sections of a canonical complex line bundle over the Grassmannian
A different injection, known as the Schneider embedding, exists for odd valuations.
-homogeneous valuations, into a certain quotient of the space of continuous sections of a line bundle over the partial flag manifold of cooriented pairs
[5] The classical theorems of Hadwiger, Schneider and McMullen give fairly explicit descriptions of valuations that are homogeneous of degree
is the Haar measure, defines a smooth even valuation of degree
The Irreducibility Theorem implies that every smooth valuation is of this form.
There are several natural operations defined on the subspace of smooth valuations
becomes a commutative associative graded algebra with the Euler characteristic as the multiplicative identity.
with a smooth boundary and strictly positive Gauss curvature are fixed, then
discovered by Alesker and enjoying many properties resembling the classical Fourier transform, which explains its name.
On even valuations, there is a simple description of the Fourier transform in terms of the Klain embedding:
For odd valuations, the description of the Fourier transform is substantially more involved.
The pushforward can be uniquely characterized by describing its action on valuations of the form
Informally, the pushforward is dual to the pullback with respect to the Alesker-Poincaré pairing: for
However, this identity has to be carefully interpreted since the pairing is only well-defined for smooth valuations.
The key observation leading to this extension is that via integration over the normal cycle (1), a smooth translation-invariant valuation may be evaluated on sets much more general than convex ones.
Also (1) suggests to define smooth valuations in general by dropping the requirement that the form
It was shown by Alesker that the smooth valuations on open subsets of
admits no natural grading in general, however it carries a canonical filtration
As in the translation-invariant case, this duality can be used to define generalized valuations.
The product of valuations closely reflects the geometric operation of intersection of subsets.
Describing the kinematic formulas explicitly is typically a difficult problem.
In fact already in the step from real to complex space forms, considerable difficulties arise and these have only recently been resolved by Bernig, Fu, and Solanes.
[12] [13] The key insight responsible for this progress is that the kinematic formulas contain the same information as the algebra of invariant valuations