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.

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Andrea Colesanti received a PhD in Mathematics at the University of Florence, where he is currently Professor in Mathematical Analysis. his main research interest is convexity, and in particular analytic aspects of convex geometry (in finite dimension). He has been interested in the connections between convex geometry and the theory of partial differential equations of elliptic type, and other areas of analysis. Together with local collaborators and colleagues from other countries, he has organized thematic workshops and schools about convex geometry. With Gabriele Bianchi and Paolo Gronchi, he has been the editor of the book Analytic Aspects of Convexity, published in the Springer INdAM.Series, in 2018. Monika Ludwig received a PhD in Mathematics at the Technische Universität Wien. She was an Erwin-Schrödinger Fellow at University College London and New York Polytechnic University and visiting professor at the University of Bern. She was a Professor at New York University before she returned to the Technische Universität Wien as a Professor of Convex and Discrete Geometry. She received the Hlawka-Prize of the Austrian Academy of Sciences and the Förderungspreis of the Austrian Mathematical Society. She became a Corresponding Member of the Austrian Academy of Sciences in 2011, a Fellow of the American Mathematical Society in 2012, and a Full Member of the Austrian Academy of Sciences in 2013. She was a plenary speaker at the European Congress of Mathematics in 2021.