Harmonic Analysis in Convex Geometry

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The problems addressed in this project concern the properties of shapes in an ambient space (like a human heart in the body) that can be inferred from the information about their shadows or slices (as in medical imaging). The beauty of the problems is that the formulations of many of them are "intuitively clear" not only to graduate students, but also to undergraduates, and in some cases, even to high-school pupils. On the other hand, the answers are very often counter-intuitive, requiring the use of the most advanced and sophisticated tools belonging to the different branches of modern mathematics. Many problems take their origin not in pure mathematics, but in medical imaging and tomography and the solutions might find very interesting biomedical applications. This project will contribute to US workforce development through the training of graduate students.The current project is a continuation of the long-time collaboration between the principal investigators. The PI and co-PI will continue to use and develop the methods of harmonic analysis to solve the problems arising in convex and discrete geometry. These problems include a new set of Bezout inequalities for mixed volumes, originating from classical Algebraic Geometry. The inequalities involve the size of projections of convex bodies and lead to very unexpected results in Information Theory and Probability. The principal investigators also plan to continue their work on the set of problems in Geometric Tomography proposed by Busemann and Petty in 1956. Only the first problem in the list has been solved so far. The PI and co-PI have made progress on a number of problems related to this list using the idea of the iteration of intersection and projection body operators. They will also continue to work on the questions related to the unique determination of convex bodies given the information on the size (or some other properties) of projections and sections.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.