Fourier Analysis in Geometric Tomography - Part 1
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Geometric tomography is the study of geometric properties of solids based on data about sections and projections of these solids. The lectures will include: 1. An outline of proofs of two of the main features of the Fourier approach to geometric tomography - the relation between the derivatives of the parallel section function of a body and the Fourier transform (in the sense of distributions) of powers of the norm generated by this body, and the Fourier characterization of intersection bodies. 2. The Busemann-Petty problem asks whether symmetric convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. We will prove an isomorphic version of the problem with a constant depending on the distance from the class of intersection bodies. This will include a generalization to arbitrary measures in place of volume. 3. The slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. We will consider a version of this problem for arbitrary measures in place of volume. We will show that the answer is affirmative for many classes of bodies, but in general the constant must be of the order 1/√n. 4. Optimal estimates for the maximal distance from a convex body to the classes of intersection bodies and the unit balls of subspaces of Lp. 5. We will use the Fourier approach to prove that the only polynomially integrable convex bodies, i.e. bodies whose parallel section function in every direction is a polynomial of the distance from the origin, are ellipsoids in odd dimensions.