Kinematic formulas for tensor valuations

Abstract

We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the Alesker-Fourier transform on the large class of spherical valuations, which is achieved by differential-geometric and representation theoretical tools. We also describe in explicit form the product and convolution of tensor valuations.

Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative

“…The tensor is equal to a linear combination of the Minkowski tensor (which is defined below, see Tables I and II) and the unit tensor multiplied by the surface area (or the perimeter for d = 2). The coefficients are explicitly given by so‐called Crofton formulas for Minkowski tensors . For local versions of these Crofton formulas, see Ref.…”

“…The coefficients are explicitly given by so-called Crofton formulas for Minkowski tensors. 41,42 For local versions of these Crofton formulas, see Ref. [43].…”

Tensor valuations on lattice polytopes

“…Tensor valuations on convex bodies have attracted increasing attention in recent years (see, e.g., [7,23,26]). They were introduced by McMullen in [37] and Alesker subsequently obtained a complete classification of continuous and isometry equivariant tensor valuations on convex bodies (based on [3] but completed in [4]).…”

SL(m,C)-equivariant and translation covariant continuous tensor valuations

“…Formula (20) follows from (19) if sin t = 0, and hence by the continuity of Z m if sin t = 0. Now Z m is smooth because it is invariant under SU(m) (see Proposition 3.6), and ϕ t ∈ GL(2m, R), t ∈ (− π 2 , π 2 ), is a C ∞ family of 2m × 2m matrices, thus Z m (ϕ t K) is a C ∞ function of t. Since Z m (ϕ t K) is differentiable at t = 0, but the right-hand-side of (20) is differentiable only if it vanishes, we conclude Kl Zm (L) = 0 by (20).…”