Semyon Alesker

Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture

Abstract. The main result of this paper is the description of translation invariant continuous valuations on convex sets. In particular, it provides an affirmative solution of P. McMullens conjecture, and in a stronger form.

Continuous rotation invariant valuations on convex sets

The notion of valuation on convex sets can be considered as a generalization of the notion of measure, which is deflned only on the class of convex compact sets. It is well-known that there are important and interesting examples of valuations on convex sets, which are not measures in the usual sense as, for example, the mixed volumes. Basic deflnitions and some classical examples are discussed in Section 2 of this paper. For more detailed information we refer to the surveys [Mc-Sch] and [Mc3]. Throughout this paper all the valuations are assumed to be continuous with respect to the Hausdorfi metric. Note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry. There exists an explicit description of translation invariant continuous valuations on 1 and 2 due to Hadwiger [H1] (the case of 2 is nontrivial). Continuous rigid motion invariant valuations on d are completely classifled by the remarkable Hadwiger theorem as linear combinations of the quermassintegrals (cf. [H2] or for a simpler proof [K]).

Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables

We recall known and establish new properties of the Dieudonne and Moore determinants of quaternionic matrices.Using these linear algebraic results we develop a basic theory of plurisubharmonic functions of quaternionic variables. Then we introduce and briefly discuss quaternionic Monge-Ampere equations.

Theory of valuations on manifolds, III. Multiplicative structure in the general case

This is the third part of a series of articles where the theory of valuations on manifolds is constructed. In the second part of this series the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from the first authors 2004 paper and, from the first part of this series.

Theory of valuations on manifolds, I. Linear spaces

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (non-translation invariant) valuations on a linear spaceV. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [4] obtained for polynomial valuations.

Range characterization of the cosine transform on higher Grassmannians

Abstract We characterize the range of the cosine transform on real Grassmannians in terms of the decomposition under the action of the special orthogonal group SO ( n ). We also give a geometric interpretation of this image in terms of valuations. In addition, we discuss the non-Archimedean analogues.

The product on smooth and generalized valuations

The product of smooth valuations on manifolds is described in terms of differential forms, Gelfand transforms, and blow-up spaces. It is shown that the product extends partially to generalized valuations and corresponds geometrically to transversal intersections. This result is used to prove a general kinematic formula on compact rank one symmetric spaces.

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry.

A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J, K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics. We prove a quaternionic analogue of A. D. Aleksandrov and ChernLevine-Nirenberg theorems.

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍn. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].

Theory of Valuations on Manifolds, IV. New Properties of the Multiplicative Structure

This is the fourth part in the series of articles math.MG/0503397, math.MG/0503399, math.MG/0509512 where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in math.MG/0503399 is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations introduced in math.MG/0503399 is an automorphism of the algebra of valuations. Then an integration functional on valuations with compact support is introduced, and a property of selfduality of valuations is proved. Next a space of generalized valuations is defined, and some basic properties of it are proved. Finally a canonical imbedding of the space of constructible functions on a real analytic manifold into the space of generalized valuations is constructed, and various structures on valuations are compared with known structures on constructible functions.