Invariant Valuations Research Articles

Valuation theory of indefinite orthogonal groups

Let SO+(p,q) denote the identity connected component of the real orthogonal group with signature (p,q). We give a complete description of the spaces of continuous and generalized translation- and SO+(p,q)-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invariant valuations. As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain–Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations. The operations of pull-back and push-forward by a linear map extend naturally to this class.

Crofton formulas and indefinite signature

We study the $O(p,q)$-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an $O(p,q)$-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the $(p-1)$ homogeneous case of $O(p,p)$, yielding a Crofton formula for the centro-affine surface area when $p\not\equiv 3\mod 4$. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all $O(p,2)$-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the $|\det X|^s$ family of distributions on the space of symmetric matrices, which we use to construct a family of $O(p,q)$-invariant Crofton distributions. We conjecture there are no others, which we then prove for $O(p,2)$ with $p$ even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.

SL(n) Invariant Valuations on Polytopes

A classification of SL$(n)$ invariant valuations on the space of convex polytopes in $R^n$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $R^n$ that contain the origin.

Integral geometry of translation invariant functionals, II: The case of general convex bodies

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures.

Spin-invariant valuations on the octonionic plane

The dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of Val2Spin (9).

Radial continuous rotation invariant valuations on star bodies

We characterize the positive radial continuous and rotation invariant valuations V defined on the star bodies of Rn as the applications on star bodies which admit an integral representation with respect to the Lebesgue measure. That is,V(K)=∫Sn−1θ(ρK)dm, where θ is a positive continuous function, ρK is the radial function associated to K and m is the Lebesgue measure on Sn−1. As a corollary, we obtain that every such valuation can be uniformly approximated on bounded sets by a linear combination of dual quermassintegrals.

Generalized translation invariant valuations and the polytope algebra

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.

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.

EXTENSIONS OF TRANSLATION INVARIANT VALUATIONS ON POLYTOPES

We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we introduce flag support measures of convex bodies via a local Steiner formula and derive some of the properties of these measures.

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1) on the real Grassmann manifolds Grk(H2) in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H2 which are invariant under the action of the group Sp(2)Sp(1).

Corrigendum to “A Fourier type transform on translation invariant valuations on convex sets”

Corrigendum to “A Fourier type transform on translation invariant valuations on convex sets”

Integral geometry of complex space forms

We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.

The Centro-Affine Hadwiger Theorem

All upper semicontinuous and $\mathrm {SL}(n)$ invariant valuations on convex bodies containing the origin in their interiors are completely classified. Each such valuation is shown to be a linear combination of the Euler characteristic, the volume, the volume of the polar body, and the recently discovered Orlicz surface areas.

Invariant Measure Under the Affine Group Over

A rational polyhedron$P\subseteq {\mathbb{R^n}}$ is a finite union of simplexes in ${\mathbb{R^n}}$ with rational vertices. P is said to be $\mathbb Z$-homeomorphic to the rational polyhedron $Q\subseteq {\mathbb{R^{\it m}}}$ if there is a piecewise linear homeomorphism η of P onto Q such that each linear piece of η and η−1 has integer coefficients. When n=m, $\mathbb Z$-homeomorphism amounts to continuous $\mathcal{G}_n$-equidissectability, where $\mathcal{G}_n=GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$ is the affine group over the integers, i.e., the group of all affinities on $\mathbb{R^{n}}$ that leave the lattice $\mathbb Z^{n}$ invariant. $\mathcal{G}_n$ yields a geometry on the set of rational polyhedra. For each d=0,1,2,. . ., we define a rational measure λd on the set of rational polyhedra, and show that any two $\mathbb Z$-homeomorphic rational polyhedra $$P\subseteq {\mathbb{R^n}}$$ and $Q\subseteq {\mathbb{R^{\it m}}}$ satisfy $\lambda_d(P)=\lambda_d(Q)$. $\lambda_n(P)$ coincides with the n-dimensional Lebesgue measure of P. If 0 ≤ dim P=d < n then λd(P)>0. For rational d-simplexes T lying in the same d-dimensional affine subspace of ${\mathbb{R^{\it n}}, $\lambda_d(T)$$ is proportional to the d-dimensional Hausdorff measure of T. We characterize λd among all unimodular invariant valuations.

The module of unitarily invariant area measures

The hermitian analog of Aleksandrov’s area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily invariant area measures is described explicitly.

Closed form of the rotational Crofton formula

AbstractThe closed form of a rotational version of the famous Crofton formula is derived. In the case where the sectioned object is a compact d‐dimensional C2 manifold with boundary, the rotational average of intrinsic volumes (total mean curvatures) measured on sections passing through a fixed point can be expressed as an integral over the boundary involving hypergeometric functions. In the more general case of a compact subset of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R}^d$\end{document} with positive reach, the rotational average also involves hypergeometric functions. For convex bodies, we show that the rotational average can be expressed as an integral with respect to a natural measure on supporting flats. It is an open question whether the rotational average of intrinsic volumes studied in the present paper can be expressed as a limit of polynomial rotation invariant valuations.

Average of the mean curvature integral in complex space forms

Abstract In real space forms, given a regular hypersurface S, it is known that the integral over the space of s-planes of the mean curvature integral of intersections with S is a multiple of the mean curvature integral of S. We study the corresponding expression in a complex space form. The role of the s-planes is played by the complex s-planes. We prove that the same property does not hold but another term appears. We express this term as the integral of the normal curvature in the direction obtained from applying the complex structure to the normal direction. As an application, we give the measure of the set of complex lines meeting a compact domain in a complex space form, and we characterize the reproductive continuous invariant valuations of degree 2n – 2 in the standard Hermitian space.

Valuations on Lp-Spaces

A representation theorem is established for continuous valuations on L p -spaces whose underlying measure is non-atomic. As a consequence, a complete classification is obtained of continuous translation invariant valuations on and of continuous rotation invariant valuations on L p (S n−1 ).2000 AMS subject classification: 46E30 (52B45)

A classification of SL(n) invariant valuations

A classification of upper semicontinuous and SL(n) invariant valuations on the space of n-dimensional convex bodies is established. As a consequence, complete characterizations of cen t ro-affine and L p affine surface areas are obtained. The proofs make use of a new SL(n) shaping process for convex bodies.

A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line

The Alesker–Poincaré pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pairing. As an application, the product structure of the space of \mathrm{SU}(2) - and translation invariant valuations on the quaternionic line is described. The principal kinematic formula on the quaternionic line ℍ is stated and proved.