Grant: Discovery Grants Program - Individual

The theory of convexity is in high demand due to its extensive use in the description of mathematical objects in science and engineering. In this proposal, the PI will continue his long-term efforts to better understand the analytic and geometric properties of convex sets and log-concave functions as well as their applications to computer science, mathematical physics, (quantum) information theory, partial differential equations, probability, and statistics. In particular, this proposal aims to expand the existing or develop new theories for convex sets and log-concave functions which are extensions or variants of the Brunn-Minkowski theory of convex bodies. One part of the proposal will continue the PI's ongoing investigations on the Orlicz-Brunn-Minkowski theory for convex bodies and its dual theory with central focus on e.g., the Minkowski type problems, the Gauss image type problems, isoperimetric inequalities, the sine polarity or c-polarity, the sine transforms, and their connections with analysis, differential geometry, partial differential equations, information theory, etc. The second part of the proposal is dedicated to the development of the Brunn-Minkowski theory for unbounded convex sets, its extensions and dual. Unbounded convex sets play crucial roles in commutative algebra, complex analysis, and singularity theory. Hence the study of properties for unbounded convex sets is crucial and in high demand. These directions are still in their infancy and provide numerous research opportunities such as the (L_p or dual) Minkowski problems. In this proposal, the PI aims to solve various Minkowski type problems for unbounded convex sets by using the variational method and techniques from PDEs. The third part of this proposal concerns the analytic lifting of convex geometry: geometrization of log-concave functions, which has attracted great attention recently. The PI's primary goals in this direction are to advance his research in the development of geometric theory of log-concave functions and further explore its connection with information theory. In particular, the PI will continue his work on the (L_p, Orlicz, dual) Minkowski problems for log-concave functions, isoperimetric inequalities, the variational formulas in terms of the f-divergence and related Minkowski type problems, and their applications to (quantum) information theory, probability, and statistics. The research topics outlined in this proposal are diverse and interdisciplinary, and likely have a broader impact in many areas. It contains projects suitable for both undergraduate and graduate students. These greatly help the PI recruit students and get young researchers involved in his research group. The PI will pay special attention to the training of under-represented members in the field of mathematics.