Analytic and geometric aspects of convexity theory with applications

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The omnipresent convexity appears naturally when describing objects of interest in many mathematical related sciences. For instance, the set of quantum states in finite dimensional quantum systems and its subset containing all separable quantum states (i.e., not entangled quantum states) are convex compact sets. Hence, understanding the analytic and/or geometric aspects of convexity theory is in great demand and is prerequisite to understanding convex objects of interest. To this end, my proposed program of research aims to study properties of convexity theory, and to apply tools from convexity theory to attack problems arising in other areas such as mathematical physics, partial differential equations, probability theory and (quantum) information theory. One part of the proposed program of research deals with the modern geometric extensions of the Brunn-Minkowski theory and its dual. The emphasis is on understanding the properties of affine invariants (e.g., affine and geominimal surface areas), establishing new affine isoperimetric and isocapacitary inequalities, and solving Minkowski type problems (e.g., the Orlicz-Minkowski problem as well as its dual and/or polar analogues). Several projects are proposed to further explore the connections of the Brunn-Minkowski theory of convex bodies and its dual with partial differential equations, with particular attention paid to geometric inequalities, (polar or dual) Minkowski type problems, and the development of a dual Brunn-Minkowski theory for various variational functionals. Another part of the proposed program of research lies in the areas of geometrization of log-concave measures (or functions) and the information theory. The geometrization of log-concave measures can be viewed as the functional analogue of the Brunn-Minkowski theory. I aim to build a framework of the functional Lp and/or Orlicz Brunn-Minkowski theories for log-concave or quasi-concave functions, extend the entropy power inequality to their Lp and/or Orlicz analogues, and discover new geometric inequalities for quantum states. It is expected that these projects help further advance the connections between information theory and the Brunn-Minkowski theory, with particular attention paid to geometric inequalities for quantum states, and the generalizations of the entropy power inequality and Fisher information (in both classical and quantum settings). I will continue my commitment to the training of (undergraduate and graduate) students and postdocs. This program of research includes multiple diverse and interdisciplinary research topics, which makes it easier to attract Highly Qualified Personnel (HQP) and helps produce knowledgeable mathematicians of next generation.