Applications of Partial Differential Equations to Convex Geometry

Abstract

These notes were written following a summer mini-course given by the author at the XXIII School of Mathematics Lluis Santaló held at the Universidad Internacional Menendez Pelayo (UIMP) in Santander, Spain, in 2024. The course comprised four one-hour lectures and introduced geometric equations, with particular emphasis on applications in convex geometry. The notes start by exploring standard parametrizations of smooth convex bodies and their connection to various curvature measures. Fundamental tools such as the inverse Gauss map parametrization and the support function are introduced, resulting in the derivation of the curvature function of a convex body in spherical coordinates. These tools are employed to discuss problems of prescribing surface areas, framed as Monge-Ampère type equations and related elliptic partial differential equations on the sphere, and to present some characterizations of ellipsoids and Euclidean balls. The last part of the course turns to curvature flows as geometric parabolic partial differential equations. These equations are presented both as techniques for establishing the existence of convex bodies with prescribed curvature properties and as methods for proving geometric inequalities. The material, intended to be accessible to students with only basic prior knowledge of differential equations, serves as an entry point to more advanced topics in geometric analysis and PDEs in convex geometry. References have been kept to a minimum, limited to those that are both essential and of broader relevance, and from which the interested reader may pursue more specialized developments