Monotone translation invariant valuations on convex bodies

Archiv der Mathematik -     

Continuous valuations on convex sets

((Without abstract)) Submitted: December 1997     

A fourier-type transform on translation-invariant valuations on convex sets

Let V be a finite-dimensional real vector space. Let V al ^sm (V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism $$ \mathbb{F}_V :Val^{sm} (V)\tilde \to Val^{sm} (V^* ) \otimes Dens(V) $$ such that $ \mathbb{F}_V $ commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.     

Dimension of the space of unitary equivariant translation invariant tensor valuations

Following the work of Semyon Alesker in the scalar valued case and of Thomas Wannerer in the vector valued case, the dimensions of the spaces of continuous translation invariant and unitary equivariant tensor valuations are computed. In addition, a basis in the vector valued case is presented.     

Rigidity of invariant convex sets in symmetric spaces

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.     

Even valuations on convex bodies

The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry.

     

On closedness of law-invariant convex sets in rearrangement invariant spaces

Archiv der Mathematik - This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $${mathcal {X}}$$. In particular, we show...     

Lattice invariant valuations on rational polytopes

Let be a lattice ind-dimensional euclidean space , and the rational vector space it generates. If is a valuation invariant under, andP is a polytope with vertices in , then for non-negative integersn there is an expression , where the coefficient(P, n) depends only on the congruence class ofn modulo the smallest positive integerk such that the affine hull of eachr-face ofk P is spanned by points of. Moreover, _r satisfies the Euler-type relation where the sum extends over all non-empty facesF ofP. The proof involves a specific representation of simple such valuations, analogous to Hadwiger's representation of weakly continuous valuations on alld-polytopes. An example of particular interest is the lattice-point enumeratorG, whereG(P) = card(P); the results of this paper confirm conjectures of Ehrhart concerningG.     

Local tensor valuations on convex polytopes

Local versions of the Minkowski tensors of convex bodies in $n$ -dimensional Euclidean space are introduced. An extension of Hadwiger’s characterization theorem for the intrinsic volumes, due to Alesker, states that the continuous, isometry covariant valuations on the space of convex bodies with values in the vector space of symmetric $p$ -tensors are linear combinations of modified Minkowski tensors. We ask for a local analogue of this characterization, and we prove a classification result for local tensor valuations on polytopes, without a continuity assumption.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ 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${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

Characterizations of solution sets of convex vector minimization problems

Complete dual characterizations of the weak and proper optimal solution sets of an infinite dimensional convex vector minimization problem are given. The results are expressed in terms of subgradients, Lagrange multipliers and epigraphs of conjugate functions. A dual condition characterizing the containment of a closed convex set, defined by a cone-convex inequality, in a reverse-convex set, plays a key role in deriving the results. Simple Lagrange multiplier characterizations of the solution sets are also derived under a regularity condition. Numerical examples are given to illustrate the significance of the results.     

Characterizations of optimal solution sets of convex infinite programs

In this paper, several Lagrange multiplier characterizations of the solution set of a convex infinite programming problem are given. Characterizations of solution sets of cone-constrained convex programs are derived as well. The procedure is then adopted to a semi-convex problem with convex constraints. For this problem, we present firstly a necessary and sufficient condition for optimality and secondly a characterization of its optimal solution set, based on a Lagrange multiplier associated with a given solution and on directional derivatives of the objective function.      

A simple characterization of solution sets of convex programs

By means of elementary arguments we first show that the gradient of the objective function of a convex program is constant on the solution set of the problem. Furthermore the solution set lies in an affine subspace orthogonal to this constant gradient, and is in fact in the intersection of this affine subspace with the feasible region. As a consequence we give a simple polyhedral characterization of the solution set of a convex quadratic program and that of a monotone linear complementarity problem. For these two problems we can also characterize a priori the boundedness of their solution sets without knowing any solution point. Finally we give an extension to non-smooth convex optimization by showing that the intersection of the subdifferentials of the objective function on the solution set is non-empty and equals the constant subdifferential of the objective function on the relative interior of the optimal solution set. In addition, the solution set lies in the intersection with the feasible region of an affine subspace orthogonal to some subgradient of the objective function at a relative interior point of the optimal solution set.     

Infinite dimensional convex sets with a property common to finite dimensional convex sets

We characterize those metrizable compact convex sets K that contain a Bauer simplex B such that any two affine continuous functions on K coinciding on B are equal; further, we show that all metrizable simplexes and dual balls of separable L¹-preduals have this property.     

Characterization of solution sets of convex optimization problems in Riemannian manifolds

Archiv der Mathematik - In this paper, a characterization of the solution sets of convex smooth optimization programmings on Riemannian manifolds, in terms of the Riemannian gradients of the cost...