Convex Geometry

This book collects the lecture notes of the Summer School on Convex Geometry, held in Cetraro, Italy, from August 30th to September 3rd, 2021. Convex geometry is a very active area in mathematics with a solid tradition and a promising future. Its main objects of study are convex bodies, that is, compact and convex subsets of n-dimensional Euclidean space. The so-called Brunn--Minkowski theory currently represents the central part of convex geometry. The Summer School provided an introduction to various aspects of convex geometry: The theory of valuations, including its recent developments concerning valuations on function spaces; geometric and analytic inequalities, including those which come from the Lp Brunn--Minkowski theory; geometric and analytic notions of duality, along with their interplay with mass transportation and concentration phenomena; symmetrizations, which provide one of the main tools to many variational problems(not only in convex geometry). Each of these parts is represented by one of the courses given during the Summer School and corresponds to one of the chapters of the present volume. The initial chapter contains some basic notions in convex geometry, which form a common background for the subsequent chapters. The material of this book is essentially self-contained and, like the Summer School, is addressed to PhD and post-doctoral students and to all researchers approaching convex geometry for the first time.