Isoperimetric Inequalities, Brunn–Minkowski Theory and Minkowski-Type Monge–Ampère Equations on the Sphere

This volume presents a rigorous study of the Brunn–Minkowski theory, the isoperimetric inequality, and their functional and analytic extensions. It develops the classical theory of convex bodies and mixed volumes, highlighting the geometric foundations of volume and surface-area inequalities. Building on this, the text examines isoperimetric problems in both the classical and measure-theoretic settings, including sets of finite perimeter, and establishes functional analogues of geometric inequalities. A significant part of the book is devoted to the interaction between these inequalities and curvature-related equations, with Monge–Ampère equations on the sphere serving as a principal example. The exposition emphasizes structural relations: how classical convex geometry informs functional inequalities, and how analytic methods extend geometric results to broader settings. Additional topics include symmetrization techniques, stability issues, and connections with variational principles, illustrating the conceptual bridges between geometry, analysis, and partial differential equations. Throughout, the presentation integrates classical results, modern extensions, and analytic tools to provide a coherent framework for understanding the interplay between volume, surface, curvature, and functional inequalities.