Valuation (geometry)
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. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space formula_2 Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. In the geometric setting, often continuity (or smoothness) conditions are 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, heavily reliant on advanced tools from abstract algebra. Definition. 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 Examples. 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 Valuations on convex bodies. 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. Homogeneous  valuations. 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. Embedding theorems. 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. Irreducibility Theorem . 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. Smooth valuations. 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. Operations on translation-invariant valuations. 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. Exterior product. 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 Product. 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. Alesker-Poincaré duality. 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. 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 Fourier transform. 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. Pullback and pushforward. 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. 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 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 Examples. 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. Filtration. 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 Product. 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. Pullback and pushforward. 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. Applications in Integral Geometry. 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. 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: