Some Variants of the Kakeya Problem

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(1) This project will focus on certain geometric questions from classical analysis which have their origins in the well-known Kakeya problem: find the minimum dimension of a subset of d-dimensional space which contains a line segment in each direction. Of particular interest to the p.i. are estimates from below for the Hausdorff dimension of sets which have large intersections with certain collections of lines, planes, or spheres (or, in higher dimensions, with certain collections of k-planes or hyperspheres). A theme of this work will be to explore the scope and limitations of a particular method, due to the p.i., for treating the averaging operators which arise in the study of these problems. (2) This project is concerned with the structure of certain complicated subsets of higher dimensional spaces. The study of these sets, while of compelling intrinsic interest, should properly be viewed as a test of our current understanding of certain objects called averaging operators. The relevant averaging operators here are of two kinds: spherical averaging operators and Radon transforms. Spherical averaging operators are basic to our understanding of the propagation of waves through fluid media such as the ocean or the atmosphere. The Radon transform and its relatives are essential, for example, in problems of tomography such as those which arise in the field of medical imaging.