Convolution of convex valuations

Abstract

We show that the natural "convolution" on the space of smooth, even, translation-invariant convex valuations on a euclidean space V, obtained by intertwining the product and the duality transform of S. Alesker J. may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valuations as well. Based on this technical result we give an application to integral geometry, generalizing Hadwiger's additive kinematic formula for SO(V) Convex Geometry, North Holland, 1993 to general compact groups G ⊂ O(V) acting transitively on the sphere: it turns out that these formulas are in a natural sense dual to the usual (intersection) kinematic formulas.

A Hadwiger-Type Theorem for the Special Unitary Group

“…Since the Fourier transform acts trivially on Val SU (n) n , the proposition follows immediately from Theorem 1.7. of [13].…”

“…This important fact is explained in [19] and [13] and used in a crucial way in the determination of k U (n) in [14].…”

Hermitian integral geometry

“…We refer to [13] Recall that a valuation φ is of degree k if φ(tK) = t k φ(K) for all t ≥ 0 and even if φ(−K) = φ(K) for all K ∈ K. The corresponding subspace of Val is denoted by Val + k . It is known [25] that the restriction of an even valuation µ of degree k to a k-dimensional subspace E ⊂ V is a multiple of the restriction of the usual Hausdorff measure vol k to E. Putting Kl µ (E) for the proportionality factor, we obtain the Klain function Kl µ ∈ C(Gr k (V )) of µ.…”

“…The algebraic approach to the kinematic formula is based on the following statement from [13]. Let p : Val G → Val G * denote the linear isomorphism induced by the Poincaré duality pairing (6)…”

Integral geometry under G2 and Spin(7)

“…Problem (i) for G = U(n) was solved by Alesker [3], problem (ii) by Fu [18]. 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).…”