Title:
Fourier Analysis in Geometric Tomography - Part 2

dc.contributor.author Koldobsky, Alexander
dc.contributor.corporatename Georgia Institute of Technology. School of Mathematics en_US
dc.contributor.corporatename University of Missouri--Columbia en_US
dc.date.accessioned 2020-01-06T20:47:11Z
dc.date.available 2020-01-06T20:47:11Z
dc.date.issued 2019-12-11
dc.description Presented on December 11, 2019 at 10:30 a.m. in the Bill Moore Student Success Center, Press Rooms A & B, Georgia Tech. en_US
dc.description Workshop in Convexity and Geometric Aspects of Harmonic Analysis en_US
dc.description Alexander Koldobsky, University of Missouri-Columbia en_US
dc.description Runtime: 58:54 minutes en_US
dc.description.abstract 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. en_US
dc.format.extent 58:54 minutes
dc.identifier.uri http://hdl.handle.net/1853/62164
dc.language.iso en_US en_US
dc.publisher Georgia Institute of Technology en_US
dc.subject Convex bodies en_US
dc.subject Sections en_US
dc.title Fourier Analysis in Geometric Tomography - Part 2 en_US
dc.type Moving Image
dc.type.genre Lecture
dspace.entity.type Publication
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
local.relation.ispartofseries Workshop in Convexity and Geometric Aspects of Harmonic Analysis
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
relation.isSeriesOfPublication c6c04653-8502-4d18-98b7-02b3b7ad9144
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