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Valuation (geometry)

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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

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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

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Some common examples of are


Let be the set of convex bodies in Then some valuations on are


Some other valuations are

  • the lattice point enumerator , where is a lattice polytope
  • cardinality, on the family of finite sets

Valuations on convex bodies

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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

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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[1] 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[2] proved that (where ) coincides with the -dimensional space of Lebesgue measures on

A valuation is simple if for all convex bodies with Schneider[3] 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

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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[2] 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[4]). 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.[3] 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.[5]

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.[5]

Irreducibility Theorem

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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[6]). 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

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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:

(1)

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

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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

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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

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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

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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

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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

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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

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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.[7]

Valuations on manifolds

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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 (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 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

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The following are examples of smooth valuations on a smooth manifold :

  • Smooth measures on
  • The Euler characteristic; this follows from the work of Chern[8] 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.[9]
  • Let be the complex projective space, and let denote the Grassmannian of all complex projective subspaces of fixed dimension The function

where the integration is with respect to the Haar probability measure on is a smooth valuation. This follows from the work of Fu.[10]

Filtration

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The space admits no natural grading in general, however it carries a canonical filtration Here consists of the smooth measures on and is given by forms in the ideal generated by where is the canonical projection.

The associated graded vector space is canonically isomorphic to the space of smooth sections where denotes the vector bundle over such that the fiber over a point is the space of -homogeneous smooth translation-invariant valuations on the tangent space

Product

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The space admits a natural product. This product is continuous, commutative, associative, compatible with the filtration: and has the Euler characteristic as the identity element. It also commutes with the restriction to embedded submanifolds, and the diffeomorphism group of acts on by algebra automorphisms.

For example, if is Riemannian, the Lipschitz-Killing valuations satisfy

The Alesker-Poincaré duality still holds. For compact it says that the pairing is non-degenerate. As in the translation-invariant case, this duality can be used to define generalized valuations. Unlike the translation-invariant case, no good definition of continuous valuations exists for valuations on manifolds.

The product of valuations closely reflects the geometric operation of intersection of subsets. Informally, consider the generalized valuation The product is given by Now one can obtain smooth valuations by averaging generalized valuations of the form more precisely is a smooth valuation if is a sufficiently large measured family of diffeomorphisms. Then one has see.[11]

Pullback and pushforward

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Every smooth immersion of smooth manifolds induces a pullback map If is an embedding, then The pullback is a morphism of filtered algebras. Every smooth proper submersion defines a pushforward map by The pushforward is compatible with the filtration as well: For general smooth maps, one can define pullback and pushforward for generalized valuations under some restrictions.

Applications in Integral Geometry

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Let be a Riemannian manifold and let be a Lie group of isometries of acting transitively on the sphere bundle Under these assumptions the space of -invariant smooth valuations on is finite-dimensional; let be a basis. Let be differentiable polyhedra in Then integrals of the form are expressible as linear combinations of with coefficients independent of and :

(2)

Formulas of this type are called kinematic formulas. Their existence in this generality was proved by Fu.[10] For the three simply connected real space forms, that is, the sphere, Euclidean space, and hyperbolic space, they go back to Blaschke, Santaló, Chern, and Federer.

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 For a precise statement, let be the kinematic operator, that is, the map determined by the kinematic formulas (2). Let denote the Alesker-Poincaré duality, which is a linear isomorphism. Finally let be the adjoint of the product map The Fundamental theorem of algebraic integral geometry relating operations on valuations to integral geometry, states that if the Poincaré duality is used to identify with then :

.

See also

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References

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  1. ^ McMullen, Peter (1980), "Continuous translation-invariant valuations on the space of compact convex sets", Archiv der Mathematik, 34 (4): 377–384, doi:10.1007/BF01224974, S2CID 122127897
  2. ^ a b Hadwiger, Hugo (1957), Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Die Grundlehren der Mathematischen Wissenschaften, vol. 93, Berlin-Göttingen-Heidelberg: Springer-Verlag, doi:10.1007/978-3-642-94702-5, ISBN 978-3-642-94703-2
  3. ^ a b Schneider, Rolf (1996), "Simple valuations on convex bodies", Mathematika, 43 (1): 32–39, doi:10.1112/S0025579300011578
  4. ^ Klain, Daniel A. (1995), "A short proof of Hadwiger's characterization theorem", Mathematika, 42 (2): 329–339, doi:10.1112/S0025579300014625
  5. ^ a b Alesker, Semyon (2018), Introduction to the theory of valuations, CBMS Regional Conference Series in Mathematics, vol. 126, Providence, RI: American Mathematical Society
  6. ^ Alesker, Semyon (2001), "Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture", Geometric and Functional Analysis, 11 (2): 244–272, doi:10.1007/PL00001675, S2CID 122986474
  7. ^ Alesker, Semyon (2011), "A Fourier-type transform on translation-invariant valuations on convex sets", Israel Journal of Mathematics, 181: 189–294, arXiv:math/0702842, doi:10.1007/s11856-011-0008-6
  8. ^ Chern, Shiing-Shen (1945), "On the curvatura integra in a Riemannian manifold", Annals of Mathematics, Second Series, 46 (4): 674–684, doi:10.2307/1969203, JSTOR 1969203, S2CID 123348816
  9. ^ Weyl, Hermann (1939), "On the Volume of Tubes", American Journal of Mathematics, 61 (2): 461–472, doi:10.2307/2371513, JSTOR 2371513
  10. ^ a b Fu, Joseph H. G. (1990), "Kinematic formulas in integral geometry", Indiana University Mathematics Journal, 39 (4): 1115–1154, doi:10.1512/iumj.1990.39.39052
  11. ^ Fu, Joseph H. G. (2016), "Intersection theory and the Alesker product", Indiana University Mathematics Journal, 65 (4): 1347–1371, arXiv:1408.4106, doi:10.1512/iumj.2016.65.5846, S2CID 119736489
  12. ^ Bernig, Andreas; Fu, Joseph H. G.; Solanes, Gil (2014), "Integral geometry of complex space forms", Geometric and Functional Analysis, 24 (2): 403–49, arXiv:1204.0604, doi:10.1007/s00039-014-0251-12
  13. ^ Bernig, Andreas; Fu, Joseph H. G. (2011), "Hermitian integral geometry", Annals of Mathematics, Second Series, 173 (2): 907–945, arXiv:0801.0711, doi:10.4007/annals.2011.173.2.7

Bibliography

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  • S. Alesker (2018). Introduction to the theory of valuations. CBMS Regional Conference Series in Mathematics, 126. American Mathematical Society, Providence, RI. ISBN 978-1-4704-4359-7.
  • S. Alesker; J. H. G. Fu (2014). Integral geometry and valuations. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser/Springer, Basel. ISBN 978-1-4704-4359-7.
  • D. A. Klain; G.-C. Rota (1997). Introduction to geometric probability. Lezioni Lincee. [Lincei Lectures]. Cambridge University Press. ISBN 0-521-59362-X.
  • R. Schneider (2014). Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, 151. Cambridge University Press, Cambridge, RI. ISBN 978-1-107-60101-7.