Harmonic Analysis in Convex Geometry

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The goal of the proposed research is to develop methods of Harmonic Analysis to solve problems from Convex Geometry including problems from Geometric Tomography and questions concerning duality and volume. Many conjectures stipulate that there must exist direct duality connections between projections and sections of convex bodies. In order to gain an understanding of these connections, it is important to try to obtain a description of the duality phenomena involved. This is where the Fourier Analysis comes into play. A major component of this proposal is to study problems, which arise naturally from the recent work of the investigators. They will use Fourier Transform and other Harmonic Analysis tools as a link between volumes of a convex body and its polar. They will also address and study other duality problems about projections and sections of convex and star bodies. Convex Geometry has a lot of real life applications. It has goals similar to those of many related and often practical areas. One of the best-known examples is Classical Computer Aided Tomography, which aims to reconstruct the density of objects by means of their line integrals. Other examples are crystallography, robotics, stereology, and electron microscopy. One of the ideas of this proposal is to connect theoretical results from Convex Geometry to those practical areas via Geometric Tomography and Harmonic Analysis. In addition, the advantage of the problems addressed in this proposal is that most of them are "intuitively clear" not only to undergraduate students, but also to children in high school or even middle school. Even the geometric notion of duality can be explained to a child in the high school. On the other hand, the answers are (very often) counter-intuitive, different in the plane and in the space, and this stimulates an interest of students to the subject. The investigators believe that those problems will help to attract more students to Harmonic Analysis and Convexity as well as to Mathematics and Science in General.