Hermitian integral geometry

Abstract

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.

Integral geometry under G2 and Spin(7)

“…If a continuous translation invariant valuation satisfies (3), then clearly its Klain function is given by W → φ(W ) 2 , which is also the Klain function of ν 3 =ν 4 . Without giving the details, we indicate that (3) follows by noting that ν 3 is a constant coefficient valuation in the sense of [13]. In fact, if N 1 (K) ∈ I 7 (V × V ) denotes Proof.…”

“…We will need some results from [13] and [10]. Let V be a hermitian vector space of complex dimension n and W ∈ Gr k (V ).…”

“…In general, the product structure of Val G determines all kinematic formulas [12], but it may be rather hard to write down explicit formulas. For G = U(n), this was recently achieved in [13], thus solving problem (iii).…”

Lagrangian Subspaces of Manifolds

“…[12,20]), so, as an application, the symplectic structure determined by the symplectic form can be used to determine the symplectic form of the Holmes-Thompson volumes restricted on lines in integral geometry of Minkowski space, about which one can see Refs. [21][22][23].…”

Volume inequalities for complex isotropic measures

“…The idea to find analogs of known results from Euclidean geometry in complex vector spaces is not new. In recent years, the study of convex bodies in C n has received considerable attention (see, e.g., [1][2][3][4][5]13,15,19,[22][23][24][25][26]33,34,37,41,42]). …”