Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations

By Semyon Alesker. 2002

144 citations recorded by the source corpus.

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/87138e86639a83cea58702b804b8a15e0073838d

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Valuations on Sobolev spaces (2012, background): “See [25, 30, 47, 48] for information on the classical theory of valuations on convex bodies and [1, 2, 3, 4, 5, 9, 16, 21, 20, 22, 23, 33, 34, 35, 38, 39, 51, 53, 54, 58] for some of the more recent results.”
Integral geometry of complex space forms (2012, background): “when G acts transitively on the unit tangent bundle SM ([4, 11]).”
Minkowski valuations intertwining the special linear group (2012, background): “, [1–4, 8, 11, 15, 20, 33, 36, 40, 41]), the study of valuations headed mainly in two different directions during the last years.”
A fourier-type transform on translation-invariant valuations on convex sets (2007, methodology): “7 was proved by the author for even valuations in [4].”
Crofton Measures and Minkowski Valuations (2010, background): “The notion of smooth translation invariant real-valued valuations was introduced by Alesker in [ 3 ].”
Divergence for s-concave and log concave functions (2013, unknown): “, [1]-[3], [9], [22], [34]- [40], [60].”
Difference bodies in complex vector spaces (2012, background): “For some references on this direction see [4, 10, 12, 13, 14].”
Kinematic formulas for tensor valuations (2014, methodology): “The picture is completed by a Fourier-type transform, due to Alesker [7, 11], which is a linear map F : Val(V ) → Val(V ∗)⊗Dens(V ) satisfying F(φ1 · φ2) = F(φ1) ∗ F(φ2) (2) and with F(2) := F ◦ F also the Plancherel-type formula (F(2)(φ))(K) = φ(−K), K ∈ K(V ).”
A Hadwiger-Type Theorem for the Special Unitary Group (2008, background): “We also establish some relations between such forms.”
Algebraic Integral Geometry (2010, methodology): “Building on previous work with Bernstein [17], Alesker first proved both versions of the Hard Lefschetz theorem in the even case [7, 8].”