Harmonic Analysis and Convexity

A central theme of this project is the study of geometric properties of convex bodies based on information about their sections and projections. This branch of convex geometry is called geometric tomography. An important instance of this theory is x-ray tomography, which has numerous applications in science, medicine and engineering. The PI has developed a new approach to geometric tomography in which the geometric properties of convex bodies are expressed in terms of integral transforms, enabling the use of certain analytical methods to solve geometric problems. In this project, the PI plans to further develop these techniques and apply them to a range of problems at the interface between convex geometry, functional analysis, harmonic analysis and probability. For example, can one find an algebraic equation whose solutions are sections of a given solid? Can one estimate the volume of a solid from data involving areas of certain sets of sections or projections of this solid? Which random variables are stable, i.e. have the property that the sums of several copies of these variables always reproduce the same variable up to a constant? The PI will continue to work with students and early career stage mathematicians, to introduce them to this evolving area of research. The problems considered in this proposal connect several areas of mathematics - convex geometry, functional analysis and probability. However, the strategy of solution is common for most of the results - the question is translated into the language of the Fourier transform and then treated as a problem from harmonic analysis. The PI plans to consider the lower-dimensional and non-symmetric versions of the Busemann-Petty problem asking whether a convex body with uniformly smaller areas of plane sections necessarily has smaller volume. Another direction is to study comparison problems and lower estimates for the Radon transform associated with volumetric results about convex bodies. The PI plans to study algebraic properties of the Radon transform related to the problem of Arnold, going back to Lemma 28 from Newton's ``Principia." The problem is to characterize those convex domains whose cut-off area is an algebraic function of the parameters of the cutting plane. A connection with functional analysis is the study of embedding and duality problems. An old problem about norm dependent positive definite functions is related to embeddings of normed spaces and stable random vectors. 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.