Hermitian integral geometry
By A. Bernig, Joseph H. G. Fu. 2008
167 citations recorded by the source corpus.
We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.
View Paper: https://www.semanticscholar.org/paper/cf1fabebdbfaadab2306af64054c990bdbe24947
How researchers see this paper
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- A. Bernig
- Franz E. Schuster
- Monika Ludwig
- Judit Abardia
- Semyon Alesker
- Andrea Colesanti
- Thomas Wannerer
- Daniel Hug
Citations in the corpus
- Minkowski valuations and generalized valuations (2015, background): “These results in turn were critical for the tremendous progress in integral geometry of recent years (see [5, 16, 17, 20, 35, 66] and the references therein).”
- Integral geometry of complex space forms (2012, methodology): “The case λ = 0 reduces to the case (M,G) = (Cn,U(n)) studied in [19].”
- Kinematic formulas for tensor valuations (2014, background): “We refer to [15, 16, 17, 18, 19] for the relation between algebraic structures on scalar valued valuations and integral-geometric formulas.”
- The Hadwiger Theorem on Convex Functions, I (2020, background): “Interesting new valuations keep arising (see, for example, [8,24,31]) and see [1,2,4,6,7,21,22,29,32,35,36,39] for some recent classification results.”
- Valuations on convex functions and convex sets and Monge–Ampère operators (2017, background): “The recent progress in understanding of the structure of such valuations allowed Bernig and Fu to give an alternative description of it in integral geometric terms (see [7], Lemma 3.”
- Divergence for s-concave and log concave functions (2013, unknown): “, [1]-[3], [9], [22], [34]- [40], [60].”
- The product on smooth and generalized valuations (2009, unknown): “For an entire function f(z) = ∑∞ k=0akz k, ak ∈ C, the power series ∞ ∑ k=0 akμ k converges to a smooth valuation denoted by f(μ).”
- Difference bodies in complex vector spaces (2012, background): “For some references on this direction see [4, 10, 12, 13, 14].”