Applications of Analytic and Probabilistic Methods in Convexity to Geometric Functionals

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The proposed research program belongs to the related fields of convex geometry, geometric functional analysis and asymptotic geometric analysis. These are very active research areas that interact with each other and that are closely connected to many other fields such as differential geometry, harmonic analysis, probability theory and combinatorics. Developments in these areas have found and continue to find applications in mathematical physics, information theory, data science, geophysics, computer science among others. One of the central elements of convexity theory is the so-called Steiner formula which states that the volume of convex bodies behaves as a polynomial. The coefficients that arise in this polynomial are classical geometric invariants, called intrinsic volumes or quermassintegrals, the study of which goes back to Hermann Minkowski. The main goal of this proposal is the study of functionals associated with convex bodies, including quermassintegrals and their generalizations, in various settings. I plan to use techniques from geometric functional analysis in combination with probabilistic tools and others to approach the problems. The proposal consists of three main parts. The first topic is related to the study of the Lp Steiner formula via affine surface area. The second part is about isoperimetric and reverse isoperimetric problems. The last part concerns stochastic counterparts of inequalities for functionals associated with convex bodies.