Shape Discovery for Convex Bodies: Measures, Invariants, and Applications

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This research project concerns constructing, identifying, or describing geometric objects from either indirect measurements or from a set of design requirements. Such construction problems arise in mathematics, science, engineering, and medicine. Identifying organs and tumors within the human body is one such example. The design of the shapes of radar antennas is another. The Brunn-Minkowski theory is at the heart of one of the core subjects that provides the mathematical and computational tools used in such applications. The core of the Brunn-Minkowski theory concerns the study of types of geometric measurements (such as the surface area, volume, curvature of its boundary) of an object. Another central area are Minkowski problems, which ask whether an object can be reconstructed from a set of these geometric measurements. The investigators will continue their efforts to expand and enrich both of these central areas. They will also continue their work connecting ideas in information theory with Brunn-Minkowski theory. The work of the investigators has generated interesting questions, some that have still withstood the efforts of the best research mathematicians and other problems that can be explored even by high school and undergraduate students.The recent discovery of the dual curvature measures has led to new compelling problems to be attacked. One of them is the dual Minkowski problem which requires solving a novel fully nonlinear degenerated partial differential equations with measure data. By combining techniques from geometry and analysis, the Investigators (with various collaborators) have been developing new methods to solve characterization problems for geometric measures that are related to degenerated partial differential equations with measure data. These newly developed techniques (of the Investigators) of obtaining delicate estimates for integrals with respect to measures have led to the discovery (by the Investigators) that "measure concentration" is the key phenomenon displayed by solutions to those geometric characterization problems. Continuing these investigations should enable the Investigators to make significant progress on fundamental problems regarding characterizing geometric measures. Affine isoperimetric inequalities have for many years been a central focus of the Investigatoirs' efforts and a number of new directions are to be explored. The study (pioneered by the Investigators) of connections between affine isoperimetric inequalities and sharp affine Sobolev inequalities shows much promise and will be further explored. The Investigators will continue to exploit connections between their quadratic Brunn-Minkowski theory and the subject of information theory 9from electrical engineering).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.