Geometric, convexity and regularity properties of certain classes of convex bodies

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This project focuses on the application of Analysis techniques, in particular the Fourier transform, to the study of the geometric and convexity properties of certain classes of convex bodies: Zonoids and intersection bodies. These bodies have appeared independently in several areas of Mathematics, from Geometry to Functional Analysis. In the last two decades, the introduction of Fourier analytic techniques has yielded important results about these two classes of bodies, and has been used to unify the study of some of their properties. However, many open questions still remain. The proposed research is directed in three main directions: (a) a better understanding of the relation between zonoids and intersection bodies, such as conditions that ensure that an intersection body is the dual of a zonoid, or the study of the asymptotic behavior of zonoids whose polars are zonoids; (b) the study of the convexity properties and fixed points of the intersection body operator; (c) reconstruction of non-symmetric bodies from lower dimensional information, such as the area or the perimeter of their parallel sections. Convex bodies appear as objects of study in many areas of Mathematics, and have applications in Physics, Medicine, and Computer Science. For example, the problem of reconstructing a three dimensional body from one and two dimensional information is the basis of medical techniques such as X-rays and CT-scans, and it is also used in the study of the structure and physical properties of crystals. Convexity holds central importance in optimization and linear programming problems. This project works towards the advancement of the mathematical understanding convex bodies and the relation between their analytic and geometric properties. The proposed educational program, which includes graduate classes and seminars, undergraduate projects at North Dakota State University, and visitor exchanges with other Universities, will help prepare mathematicians to work in these areas. Further outreach activities, such as Sonia Kovalevsky Day for high school girls and the NDSU Math Club, will engage underrepresented students in the study of Mathematics.