Harmonic Analysis in Convex Geometry

(in alphabetic order by speaker surname) Speaker: Judit Abardia (Frankfurt University) Title: Projection bodies in complex vector spaces Abstract: The projection body of a convex body in the Euclidean space was characterized by Monika Ludwig as the unique Minkowski valuation which is continuous, translation invariant and contravariant under the (real) special linear group. In a joint work with Andreas Bernig, we study the complex analog of this characterization result, i.e. we describe the space of Minkowski valuations on an n-dimensional complex vector space which are continuous, translation invariant and contravariant under the complex special linear group. A family of valuations satisfying these properties appear. We show that each of these valuations satisfy geometric inequalities of Brunn-Minkowski, Alexandrov-Fenchel and Minkowski type. Speaker: David Alonso-Gutiérrez (University of Alberta) Title: On the factorization of Sobolev inequalities through classes of functions Abstract: We recall two approaches to recent improvements of the classical Sobolev inequality and give a relation between them. (Joint work with J. Bastero and J. Bernués) Speaker: Gautier Berck (Polytechnic Institute of NYU) Title: Regularized Integral Geometry Abstract: Divergent integrals may appear in classical integral geometry due to the lack of compacity of certain isotropy subgroups. We will show how to get round this difficulty in specific situations replacing the invariant measures by invariant distributions. Speaker: Paul Goodey (University of Oklahoma) Title: Harmonic analysis aspects of local and equatorial determination Abstract: We will discuss the harmonic analysis that underlies a number of recent results concerning the local or equatorial determination of various classes of bodies, both symmetric and non-symmetric. We will also explain how these results are related to certain support properties of functional operators associated with these geometric objects. Speaker: Yehoram Gordon (Technion) Title: Some Geometric functional inequalities Speaker: Eric Grinberg (University of Massachusetts, Boston) Title: Tomography in Affine and Projective Geometries over finite fields Abstract: In the standard mathematical model of tomography, an unknown function in euclidean space is to be recovered from data regarding its integrals over certain families of lines, planes, etc. The treatment of this problem involves both the geometry of the collection of lines, planes etc., and the analysis of function spaces that model the data. Here we replace euclidean space by an affine or projective space over a finite field, so as to focus the recovery and inversion problem on the collection lines involved. We give a series