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.