Theory of Valuations on Manifolds: A Survey
By Semyon Alesker. 2006
88 citations recorded by the source corpus.
Abstract.This is a non-technical survey of a recent theory of valuations on manifolds constructed in [A10], [A11], [AF] and [A12], and actually a guide to this series of articles. We also review some recent related results obtained by a number of people.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/be0f41995250b8e3b7c6a76360c2d84a0a9e63de
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In Applied Mathematics
- A. Bernig
- Judit Abardia
- Semyon Alesker
- Franz E. Schuster
- Monika Ludwig
- Thomas Wannerer
- Christoph Haberl
- Daniel Hug
In Geometry and Topology
- Judit Abardia
- A. Bernig
- Monika Ludwig
- Semyon Alesker
- Franz E. Schuster
- Thomas Wannerer
- Christoph Haberl
- Joseph H. G. Fu
Citations in the corpus
- Difference bodies in complex vector spaces (2012, background): “Then, we always get a finite dimensional real vector space (see [6]).”
- Fisher information and matrix-valued valuations (2011, background): “Valuations were also investigated on star shaped sets [25,26] and on manifolds [6–8,10].”
- Algebraic Integral Geometry (2010, methodology): “The theory of valuations on manifolds was recently worked out, mainly by Alesker [11, 12, 18, 14, 13, 24, 23, 16].”
- Minkowski Areas and Valuations (2010, background): “of elementary mixed volumes, valuations have become a critical notion (see [27, 31, 47] and see [1, 2, 3, 4, 5, 6, 11, 15, 29, 30, 49, 51, 52, 53] for some of the more recent contributions).”
- 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 more general context of arbitrary manifolds [8]-[10], [12] (see also a survey [11] of these results).”
- Valuations on Sobolev spaces (2012, background): “Valuations were also investigated on star shaped sets [27, 28], on manifolds [6–8, 10, 11] and on Lebesgue spaces [37,56,57].”
- Projection bodies in complex vector spaces (2011, background): “s a dense subspace in Val. Clearly, if µ ∈ Valsm,+ k , then the Klain function of µ is smooth as a function on Grk V. We will need the following fact about smooth valuations. Proposition 2.2 (Alesker [5]). If G is a subgroup of SO(V) acting transitively on the unit sphere of V, then the subspace Val”