In geometry, a valuation is a finitely additive function from a collection of subsets of a set to an abelian semigroup. 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 are surface area, mean width, and Euler characteristic.
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.
Definition Let be a set, and let be a collection of subsets of A function on with values in an abelian semigroup is called a valuation if it satisfies
whenever and are elements of If then one always assumes
Examples Some common examples of are
the convex bodies in compact convex polytopes in convex cones smooth compact polyhedra in a smooth manifold
Let be the set of convex bodies in Then some valuations on are
the Euler characteristic Lebesgue measure restricted to intrinsic volume (and, more generally, mixed volume) the map where is the support function of
Some other valuations are the lattice point enumerator , where is a lattice polytope cardinality, on the family of finite sets
Valuations on convex bodies From here on, let , let be the set of convex bodies in , and let be a valuation on .
We say is translation invariant if, for all and , we have .
Let . The Hausdorff distance is defined as
where is the -neighborhood of under some Euclidean inner product. Equipped with this metric, is a locally compact space.
The space of continuous, translation-invariant valuations from to is denoted by
The topology on is the topology of uniform convergence on compact subsets of Equipped with the norm
where is a bounded subset with nonempty interior, is a Banach space.
Homogeneous valuations A translation-invariant continuous valuation is said to be -homogeneous if
for all and The subset of -homogeneous valuations is a vector subspace of McMullen's decomposition theorem states that
In particular, the degree of a homogeneous valuation is always an integer between and
Valuations are not only graded by the degree of homogeneity, but also by the parity with respect to the reflection through the origin, namely
where with if and only if for all convex bodies The elements of and are said to be even and odd, respectively.
It is a simple fact that is -dimensional and spanned by the Euler characteristic that is, consists of the constant valuations on
In 1957 Hadwiger proved that (where ) coincides with the -dimensional space of Lebesgue measures on
A valuation is simple if for all convex bodies with Schneider in 1996 described all simple valuations on : they are given by where is an arbitrary odd function on the unit sphere and is the surface area measure of In particular, any simple valuation is the sum of an - and an -homogeneous valuation. This in turn implies that an -homogeneous valuation is uniquely determined by its restrictions to all -dimensional subspaces.
Embedding theorems The Klain embedding is a linear injection of the space of even -homogeneous valuations, into the space of continuous sections of a canonical complex line bundle over the Grassmannian of -dimensional linear subspaces of Its construction is based on Hadwiger's characterization of -homogeneous valuations. If and then the restriction is an element and by Hadwiger's theorem it is a Lebesgue measure. Hence
defines a continuous section of the line bundle over with fiber over equal to the -dimensional space of densities (Lebesgue measures) on
Theorem (Klain). The linear map is injective.
A different injection, known as the Schneider embedding, exists for odd valuations. It is based on Schneider's description of simple valuations. It is a linear injection of the space of odd -homogeneous valuations, into a certain quotient of the space of continuous sections of a line bundle over the partial flag manifold of cooriented pairs Its definition is reminiscent of the Klain embedding, but more involved. Details can be found in.
The Goodey-Weil embedding is a linear injection of into the space of distributions on the -fold product of the -dimensional sphere. It is nothing but the Schwartz kernel of a natural polarization that any admits, namely as a functional on the -fold product of the latter space of functions having the geometric meaning of differences of support functions of smooth convex bodies. For details, see.
Irreducibility Theorem The classical theorems of Hadwiger, Schneider and McMullen give fairly explicit descriptions of valuations that are homogeneous of degree and But for degrees very little was known before the turn of the 21st century. McMullen's conjecture is the statement that the valuations
span a dense subspace of McMullen's conjecture was confirmed by Alesker in a much stronger form, which became known as the Irreducibility Theorem:
Theorem (Alesker). For every the natural action of on the spaces and is irreducible.
Here the action of the general linear group on is given by
The proof of the Irreducibility Theorem is based on the embedding theorems of the previous section and Beilinson-Bernstein localization.
Smooth valuations A valuation is called smooth if the map from to is smooth. In other words, is smooth if and only if is a smooth vector of the natural representation of on The space of smooth valuations is dense in ; it comes equipped with a natural Fréchet-space topology, which is finer than the one induced from
For every (complex-valued) smooth function on
where denotes the orthogonal projection and is the Haar measure, defines a smooth even valuation of degree It follows from the Irreducibility Theorem, in combination with the Casselman-Wallach theorem, that any smooth even valuation can be represented in this way. Such a representation is sometimes called a Crofton formula.
For any (complex-valued) smooth differential form that is invariant under all the translations and every number integration over the normal cycle defines a smooth valuation:
As a set, the normal cycle consists of the outward unit normals to The Irreducibility Theorem implies that every smooth valuation is of this form.
Operations on translation-invariant valuations There are several natural operations defined on the subspace of smooth valuations The most important one is the product of two smooth valuations. Together with pullback and pushforward, this operation extends to valuations on manifolds.
Exterior product Let be finite-dimensional real vector spaces. There exists a bilinear map, called the exterior product,
which is uniquely characterized by the following two properties: it is continuous with respect to the usual topologies on and if and where and are convex bodies with smooth boundary and strictly positive Gauss curvature, and and are densities on and then
Product The product of two smooth valuations is defined by
where is the diagonal embedding. The product is a continuous map
Equipped with this product, becomes a commutative associative graded algebra with the Euler characteristic as the multiplicative identity.
Alesker-Poincaré duality By a theorem of Alesker, the restriction of the product is a non-degenerate pairing. This motivates the definition of the -homogeneous generalized valuation, denoted as topologized with the weak topology. By the Alesker-Poincaré duality, there is a natural dense inclusion
Convolution Convolution is a natural product on For simplicity, we fix a density on to trivialize the second factor. Define for fixed with smooth boundary and strictly positive Gauss curvature There is then a unique extension by continuity to a map called the convolution. Unlike the product, convolution respects the co-grading, namely if then
For instance, let denote the mixed volume of the convex bodies If convex bodies in with a smooth boundary and strictly positive Gauss curvature are fixed, then
defines a smooth valuation of degree The convolution two such valuations is
where is a constant depending only on
Fourier transform The Alesker-Fourier transform is a natural, -equivariant isomorphism of complex-valued valuations discovered by Alesker and enjoying many properties resembling the classical Fourier transform, which explains its name.
It reverses the grading, namely and intertwines the product and the convolution:
Fixing for simplicity a Euclidean structure to identify we have the identity On even valuations, there is a simple description of the Fourier transform in terms of the Klain embedding: In particular, even real-valued valuations remain real-valued after the Fourier transform.
For odd valuations, the description of the Fourier transform is substantially more involved. Unlike the even case, it is no longer of purely geometric nature. For instance, the space of real-valued odd valuations is not preserved.
Pullback and pushforward Given a linear map there are induced operations of pullback and pushforward The pullback is the simpler of the two, given by It evidently preserves the parity and degree of homogeneity of a valuation. Note that the pullback does not preserve smoothness when is not injective.
The pushforward is harder to define formally. For simplicity, fix Lebesgue measures on and The pushforward can be uniquely characterized by describing its action on valuations of the form for all and then extended by continuity to all valuations using the Irreducibility Theorem. For a surjective map For an inclusion choose a splitting Then
Informally, the pushforward is dual to the pullback with respect to the Alesker-Poincaré pairing: for and
However, this identity has to be carefully interpreted since the pairing is only well-defined for smooth valuations. For further details, see.
Valuations on manifolds In a series of papers beginning in 2006, Alesker laid down the foundations for a theory of valuations on manifolds that extends the theory of valuations on convex bodies. The key observation leading to this extension is that via integration over the normal cycle (), a smooth translation-invariant valuation may be evaluated on sets much more general than convex ones. Also () suggests to define smooth valuations in general by dropping the requirement that the form be translation-invariant and by replacing the translation-invariant Lebesgue measure with an arbitrary smooth measure.
Let be an n-dimensional smooth manifold and let be the co-sphere bundle of that is, the oriented projectivization of the cotangent bundle. Let denote the collection of compact differentiable polyhedra in The normal cycle of which consists of the outward co-normals to is naturally a Lipschitz submanifold of dimension
For ease of presentation we henceforth assume that is oriented, even though the concept of smooth valuations in fact does not depend on orientability. The space of smooth valuations on consists of functions of the form
where and can be arbitrary. It was shown by Alesker that the smooth valuations on open subsets of form a soft sheaf over
Examples The following are examples of smooth valuations on a smooth manifold : Smooth measures on The Euler characteristic; this follows from the work of Chern on the Gauss-Bonnet theorem, where such and were constructed to represent the Euler characteristic. In particular, is then the Chern-Gauss-Bonnet integrand, which is the Pfaffian of the Riemannian curvature tensor. If is Riemannian, then the Lipschitz-Killing valuations or intrinsic volumes are smooth valuations. If is any isometric immersion into a Euclidean space, then where denotes the usual intrinsic volumes on (see below for the definition of the pullback). The existence of these valuations is the essence of Weyl's tube formula.