Theory of valuations on manifolds, I. Linear spaces

By Semyon Alesker. 2005

80 citations recorded by the source corpus.

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (non-translation invariant) valuations on a linear spaceV. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [4] obtained for polynomial valuations.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2f48ca73f22cef5053ca38e9dbf78aa24d138817

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Citations in the corpus

The product on smooth and generalized valuations (2009, methodology): “hen for µ1 ∈ V∞(X),µ2 ∈ V∞(Y) and transversal P∈ P(X′ ×Y′) we have (µ1 ⊠µ2)((φ×ψ)(P)) = (φ∗µ1 ⊠ψ∗µ2)(P). (1) (2) (Aﬃne case) If Xand Yare aﬃne spaces, then µ1⊠µ2 is the exterior product introduced in [3]. The product of two valuations on X is the restriction of the exterior product on the diagonal i”
Theory of valuations on manifolds, II (2005, unknown): “Obviously SV (V ) = W0 ⊃W1 ⊃ . . . .”
Integral geometry of complex space forms (2012, background): “0 then gQ⊂ U. Finally, since the assignment U ↦ V∞(U) is a sheaf ([7], Theorem 2.4.10), it is enough to show that φ⋅ψ(P) = ∫G φ(P∩gQ)ρ(g)dgfor P∈ P(U). Let F∶ U→ Rn be the coordinate map. By [12] and [6], Thm. 5.2.2, the topological algebra of smooth valuations on Msupported in Uis isomorphic to the”
Theory of Valuations on Manifolds, IV. New Properties of the Multiplicative Structure (2005, methodology): “We refer to [ A4 ] for an axiomatic definition.”
A fourier-type transform on translation-invariant valuations on convex sets (2007, background): “Moreover, some parts of the classical theory of valuations on affine spaces have been generalized to the more general context of arbitrary manifolds [ 8 ]–[10], [12] (see also a survey [11] of these results).”
Kinematic formulas for tensor valuations (2014, background): “s called smooth if there exist translation invariant diﬀerential forms γ∈ Ωn−1(SV) and τ ∈ Ωn(V) such that φ(K) = Z nc(K) γ+ Z K τ. The space Valsm(V) of smooth valuations is dense in Val(V), compare [8]. Alesker [7] has shown that Valsm(V) has the structure of an associative, commutative graded alg”
Harmonic Analysis of Translation Invariant Valuations (2010, methodology): “differential forms of bidegree (i, n − i − 1) on SV . The following result is a special case of [ Al6 , Th. 5.2.1]: Lemma 2.3.”
Hermitian integral geometry (2008, background): “However, many statements apply also to other geometrically valid subsets (e.g. C2 submanifolds, or in the case of the Crofton formulas even C1 submanifolds) of smooth manifolds, in terms of the formalism of valuations on manifolds [5, 6, 9, 7, 8].”
Algebraic Integral Geometry (2010, methodology): “o valuations and not to curvature measures. (2) We have described in detail the theory of valuations on an aﬃne space. The theory of valuations on manifolds was recently worked out, mainly by Alesker [11, 12, 18, 14, 13, 24, 23, 16]. This gives the appropriate framework to study integral geometry of”