[Paper Review] Integral geometric formulae for Minkowski tensors

Daniel Hug, Jan A. Weis|arXiv (Cornell University)|Dec 27, 2017

Point processes and geometric inequalities19 references3 citations

TL;DR

This paper establishes complete sets of integral geometric formulae—kinematic and Crofton formulae—for Minkowski tensors, the tensor-valued generalizations of intrinsic volumes. Using tensorial curvature measures and their known integral formulae, the authors derive explicit linear combinations expressing the average Minkowski tensors of the intersection of a convex body with a randomly moved convex body (kinematic) or affine subspace (Crofton), in terms of the original tensors, thus generalizing classical results to the tensor-valued setting.

ABSTRACT

The Minkowski tensors are the natural tensor-valued generalizations of the intrinsic volumes of convex bodies. We prove two complete sets of integral geometric formulae, so called kinematic and Crofton formulae, for these Minkowski tensors. These formulae express the integral mean of the Minkowski tensors of the intersection of a given convex body with a second geometric object (another convex body in the kinematic case and an affine subspace in the Crofton case) which is uniformly moved by a proper rigid motion, in terms of linear combinations of the Minkowski tensors of the given geometric objects.

Motivation & Objective

• To extend classical integral geometry from scalar intrinsic volumes to tensor-valued Minkowski tensors.
• To derive complete sets of kinematic and Crofton formulae for Minkowski tensors in Euclidean space.
• To establish these formulae using the theory of tensorial curvature measures and their known integral identities.
• To generalize Hadwiger’s characterization and integral geometric theorems to the tensor-valued setting.

Proposed method

• The authors use generalized tensorial curvature measures as intermediate objects, which are local, tensor-valued generalizations of curvature measures.
• They apply known kinematic and Crofton formulae for tensorial curvature measures (from prior work) as foundational tools.
• The derivation proceeds by integrating the Minkowski tensors of the intersection $ K \cap gK' $ over the group of proper rigid motions $ G_n $, using the invariance of the Haar measure.
• The key step involves expressing the average of $ \Phi_j^{r,s}(K \cap gK') $ as a linear combination of products of $ \Phi_k^{r,s'}(K) $ and $ \Phi_{n-k+j}^{r,s''}(K') $, with coefficients involving Gamma functions and geometric constants.
• For the Crofton case, integration is performed over the affine Grassmannian $ A(n,k) $, and the formulae are derived by distinguishing cases based on the dimension of the intersection.
• The final formulae are obtained by taking total measures of the tensorial curvature measures, which recover the Minkowski tensors.

Experimental results

Research questions

• RQ1 Can kinematic formulae be generalized from scalar intrinsic volumes to tensor-valued Minkowski tensors?
• RQ2 Can Crofton-type formulae be established for Minkowski tensors of convex bodies intersected with random affine subspaces?
• RQ3 What are the explicit linear coefficients relating the expected Minkowski tensors of intersections to the original tensors of the bodies involved?
• RQ4 How do the symmetries and invariance properties of the motion group and affine Grassmannian manifest in the structure of these tensor-valued formulae?
• RQ5 What is the role of tensorial curvature measures in enabling the derivation of such formulae?

Key findings

• The paper establishes a complete kinematic formula for Minkowski tensors, expressing the average of $ \Phi_j^{r,s}(K \cap gK') $ over all rigid motions $ g \in G_n $ as a finite linear combination of products of Minkowski tensors of $ K $ and $ K' $, with coefficients involving Gamma functions and geometric constants.
• A corresponding Crofton formula is derived, showing that the average of $ \Phi_j^{r,s}(K \cap E) $ over all $ k $-flats $ E \in A(n,k) $ equals a linear combination of Minkowski tensors of $ K $, with coefficients depending on dimension and tensor rank.
• The coefficients in both formulae are explicitly given in terms of the Gamma function, the dimension $ n $, and tensorial indices $ r,s,j,k $, generalizing the classical $ \alpha_{n,j,k} $ constant.
• The formulae are valid not only for convex bodies but also for polyconvex sets, due to the additivity of Minkowski tensors.
• The derivation relies crucially on the known integral formulae for tensorial curvature measures, which are used as a bridge to the final results.
• The results confirm that Minkowski tensors satisfy nontrivial integral geometric laws analogous to the scalar intrinsic volumes, extending the classical framework of integral geometry to higher-rank tensor-valued valuations.