Valuation (geometry)
Let formula_1 be a set and formula_4 be a collection of admissible subsets of formula_5 A function formula_6 on formula_4 with values in an abelian semigroup formula_8 is called a valuation if it satisfies 
formula_9
whenever formula_10 formula_11 formula_12 and formula_13 are elements of formula_14 
If formula_15 then one always assumes formula_16 Some common examples for the admissible sets formula_4 are non-empty compact convex sets (convex bodies) in formula_18 compact convex polytopes in formula_18 convex cones, and smooth compact polyhedra in a smooth manifold formula_5
Let formula_21 be a finite-dimensional vector space over formula_22 and let formula_23 denote the set of convex bodies in formula_24

formula_29
where formula_30 is a normalizing constant and formula_31 is the Euclidean unit ball, and more generally the mixed volumes (with some entries fixed arbitrarily). formula_35 
is the support function of formula_36 is a valuation on formula_28 A valuation on formula_23 is said to be "translation invariant" if formula_39 for all formula_40 and all convex bodies formula_41
Let formula_42 be two convex bodies in formula_24 After a choice of Euclidean inner product, their Hausdorff distance formula_44 is defined by
formula_45
where formula_46 denotes the formula_47-neighborhood of formula_41 Equipped with this metric, formula_23 is a locally compact space. The space of continuous, translation-invariant valuations on formula_23 taking values in the complex numbers formula_51 is denoted by formula_52
The topology on formula_53 is the topology of uniform convergence on compact subsets of formula_54 Equipped with the norm 
formula_55 
where formula_56 is a bounded subset with nonempty interior, formula_53 is a Banach space. A translation-invariant continuous valuation formula_58 is said to be "formula_59-homogeneous" if 
formula_60
for all formula_61 and formula_62 The subset formula_63 of formula_59-homogeneous valuations is a vector subspace of formula_52 McMullen's decomposition theorem states that
formula_66
In particular, the degree of a homogeneous valuation is always an integer between formula_67 and formula_68
Valuations are not only graded by the degree of homogeneity, but also by the parity with respect to the reflection through the origin, namely
formula_69 
where formula_70 with formula_71 if and only if formula_72 for all convex bodies formula_41
The elements of formula_74 and formula_75 are said to be "even" and "odd", respectively. It is a simple fact that formula_76 is formula_77-dimensional and spanned by the Euler characteristic formula_78 that is, consists of the constant valuations on formula_28
In 1957 Hadwiger proved that formula_80 (where formula_81) coincides with the formula_77-dimensional space of Lebesgue measures on formula_24
A valuation formula_84 is "simple" if formula_85 for all convex bodies with formula_86 Schneider in 1996 described all simple valuations on formula_87: they are given by formula_88 where formula_89 formula_90 is an arbitrary odd function on the unit sphere formula_91 and formula_92 is the surface area measure of formula_41 In particular, any simple valuation is the sum of an formula_94- and an formula_95-homogeneous valuation. This in turn implies that an formula_59-homogeneous valuation is uniquely determined by its restrictions to all formula_97-dimensional subspaces. The Klain embedding is a linear injection of formula_98 the space of even formula_59-homogeneous valuations, into the space of continuous sections of a canonical complex line bundle over the Grassmannian formula_100 of formula_59-dimensional linear subspaces of formula_24 Its construction is based on Hadwiger's characterization of formula_94-homogeneous valuations. If formula_104 and formula_105 then the restriction formula_106 is an element formula_107 and by Hadwiger's theorem it is a Lebesgue measure. Hence
formula_108
defines a continuous section of the line bundle over formula_100 with fiber over formula_110 equal to the formula_77-dimensional space formula_112 of densities (Lebesgue measures) on formula_113
Theorem (Klain). The linear map formula_114 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 formula_115 the space of odd formula_59-homogeneous valuations, into a certain quotient of the space of continuous sections of a line bundle over the partial flag manifold of cooriented pairs formula_117 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 formula_118 into the space of distributions on the formula_59-fold product of the formula_95-dimensional sphere. It is nothing but the Schwartz kernel of a natural polarization that any formula_121 admits, namely as a functional on the formula_122-fold product of formula_123 the latter space of functions having the geometric meaning of differences of support functions of smooth convex bodies. For details, see. The classical theorems of Hadwiger, Schneider and McMullen give fairly explicit descriptions of valuations that are homogeneous of degree formula_124 formula_125 and formula_68 But for degrees formula_127
span a dense subspace of formula_52 McMullen's conjecture was confirmed by Alesker in a much stronger form, which became known as the Irreducibility Theorem:
Theorem (Alesker). For every formula_129 the natural action of formula_130 on the spaces formula_131 and formula_132 is irreducible. Here the action of the general linear group formula_130 on formula_53 is given by 
formula_135
The proof of the Irreducibility Theorem is based on the embedding theorems of the previous section and Beilinson-Bernstein localization. A valuation formula_136 is called "smooth" if the map formula_137 from formula_130 to formula_53 is smooth. In other words, formula_6 is smooth if and only if formula_6 is a smooth vector of the natural representation of formula_130 on formula_52 The space of smooth valuations formula_144 is dense in formula_53; it comes equipped with a natural Fréchet-space topology, which is finer than the one induced from formula_52
For every (complex-valued) smooth function formula_147 on formula_148
formula_149
where formula_150 denotes the orthogonal projection and formula_151 is the Haar measure, defines a smooth even valuation of degree formula_152 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 formula_153 that is invariant under all the translations formula_154 and every number formula_155 integration over the "normal cycle" defines a smooth valuation:
As a set, the normal cycle formula_156 consists of the outward unit normals to formula_41 The Irreducibility Theorem implies that every smooth valuation is of this form. There are several natural operations defined on the subspace of smooth valuations formula_158 The most important one is the product of two smooth valuations. Together with pullback and pushforward, this operation extends to valuations on manifolds. Let formula_159 be finite-dimensional real vector spaces. There exists a bilinear map, called the exterior product, 
formula_160
which is uniquely characterized by the following two properties: 

formula_171 The product of two smooth valuations formula_172 is defined by 
formula_173
where formula_174 is the diagonal embedding. The product is a continuous map 
formula_175
Equipped with this product, formula_176 becomes a commutative associative graded algebra with the Euler characteristic as the multiplicative identity. By a theorem of Alesker, the restriction of the product 
formula_177 is a non-degenerate pairing. This motivates the definition of the formula_122-homogeneous "generalized valuation", denoted formula_179 as formula_180 topologized with the weak topology. By the Alesker-Poincaré duality, there is a natural dense inclusion formula_181 Convolution is a natural product on formula_182 For simplicity, we fix a density formula_183 on formula_21 to trivialize the second factor. Define for fixed formula_185 with smooth boundary and strictly positive Gauss curvature 
formula_186 There is then a unique extension by continuity to a map 
formula_187 called the convolution. Unlike the product, convolution respects the co-grading, namely if formula_188 formula_189 then formula_190
For instance, let formula_191 denote the mixed volume of the convex bodies formula_192 If convex bodies formula_193 in formula_87 with a smooth boundary and strictly positive Gauss curvature are fixed, then 
formula_195
defines a smooth valuation of degree formula_152 The convolution two such valuations is
formula_197
where formula_198 is a constant depending only on formula_199 The Alesker-Fourier transform is a natural, formula_130-equivariant isomorphism of complex-valued valuations 
formula_201 discovered by Alesker and enjoying many properties resembling the classical Fourier transform, which explains its name. It reverses the grading, namely formula_202 and intertwines the product and the convolution: formula_203
Fixing for simplicity a Euclidean structure to identify formula_204 formula_205 we have the identity formula_206 On even valuations, there is a simple description of the Fourier transform in terms of the Klain embedding: formula_207 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. Given a linear map formula_208 there are induced operations of pullback formula_209 and pushforward formula_210 The pullback is the simpler of the two, given by formula_211 It evidently preserves the parity and degree of homogeneity of a valuation. Note that the pullback does not preserve smoothness when formula_147 is not injective. The pushforward is harder to define formally. For simplicity, fix Lebesgue measures on formula_213 and formula_24 The pushforward can be uniquely characterized by describing its action on valuations of the form formula_215 for all formula_216 and then extended by continuity to all valuations using the Irreducibility Theorem. For a surjective map formula_217 
formula_218 For an inclusion formula_219 choose a splitting formula_220 Then 
formula_221
Informally, the pushforward is dual to the pullback with respect to the Alesker-Poincaré pairing: for formula_58 and formula_223 
formula_224
However, this identity has to be carefully interpreted since the pairing is only well-defined for smooth valuations. For further details, see. 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 formula_225 be translation-invariant and by replacing the translation-invariant Lebesgue measure with an arbitrary smooth measure. Let formula_1 be an n-dimensional smooth manifold and let formula_227 be the co-sphere bundle of formula_228 that is, the oriented projectivization of the cotangent bundle. Let formula_229 denote the collection of compact differentiable polyhedra in formula_5 The normal cycle formula_231 of formula_232 which consists of the outward co-normals to formula_10 is naturally a Lipschitz submanifold of dimension formula_234
For ease of presentation we henceforth assume that formula_1 is oriented, even though the concept of smooth valuations in fact does not depend on orientability. The space of smooth valuations formula_236 on formula_1 consists of functions formula_238 of the form
formula_239
where formula_240 and formula_241 can be arbitrary. It was shown by Alesker that the smooth valuations on open subsets of formula_1 form a soft sheaf over formula_5 The following are examples of smooth valuations on a smooth manifold formula_1:


formula_258 where the integration is with respect to the Haar probability measure on formula_259 is a smooth valuation. This follows from the work of Fu. The space formula_236 admits no natural grading in general, however it carries a canonical filtration 
formula_261 Here formula_262 consists of the smooth measures on formula_228 and formula_264 is given by forms formula_225 in the ideal generated by formula_266 where formula_267 is the canonical projection. The associated graded vector space formula_268 is canonically isomorphic to the space of smooth sections formula_269 where formula_270 denotes the vector bundle over formula_1 such that the fiber over a point formula_272 is formula_273 the space of formula_59-homogeneous smooth translation-invariant valuations on the tangent space formula_275 The space formula_236 admits a natural product. This product is continuous, commutative, associative, compatible with the filtration:
formula_277
and has the Euler characteristic as the identity element. It also commutes with the restriction to embedded submanifolds, and the diffeomorphism group of formula_1 acts on formula_236 by algebra automorphisms. For example, if formula_1 is Riemannian, the Lipschitz-Killing valuations satisfy 
formula_281
The Alesker-Poincaré duality still holds. For compact formula_1 it says that the pairing formula_283 formula_284 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 formula_285 The product is given by formula_286 Now one can obtain smooth valuations by averaging generalized valuations of the form formula_287 more precisely formula_288 is a smooth valuation if formula_289 is a sufficiently large measured family of diffeomorphisms. Then one has 
formula_290
see. Every smooth immersion formula_291 of smooth manifolds induces a pullback map formula_292 If formula_147 is an embedding, then 
formula_294
The pullback is a morphism of filtered algebras. Every smooth proper submersion formula_295 defines a pushforward map formula_296 by 
formula_297 The pushforward is compatible with the filtration as well: formula_298
For general smooth maps, one can define pullback and pushforward for generalized valuations under some restrictions. Let formula_299 be a Riemannian manifold and let formula_300 be a Lie group of isometries of formula_299 acting transitively on the sphere bundle formula_302 Under these assumptions the space formula_303 of formula_300-invariant smooth valuations on formula_299 is finite-dimensional; let formula_306 be a basis. Let formula_307 be differentiable polyhedra in formula_308 Then integrals of the form formula_309 are expressible as linear combinations of formula_310 with coefficients formula_311 independent of formula_312 and formula_313: 
Formulas of this type are called "kinematic formulas". Their existence in this generality was proved by Fu. 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. The key insight responsible for this progress is that the kinematic formulas contain the same information as the algebra of invariant valuations formula_314 For a precise statement, let formula_315
be the kinematic operator, that is, the map determined by the kinematic formulas (). Let 
formula_316
denote the Alesker-Poincaré duality, which is a linear isomorphism. Finally let formula_317 be the adjoint of the product map formula_318 The Fundamental theorem of algebraic integral geometry relating operations on valuations to integral geometry, states that if the Poincaré duality is used to identify formula_319 with formula_320 then formula_321:
Definition
In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set formula_1 with values in an abelian semigroup.