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This book presents the proceedings of the international conference Analytic Aspects in Convexity, which was held in Rome in October 2016. It offers a collection of selected articles, written by some of the world’s leading experts in the field of Convex Geometry, on recent developments in this area: theory of valuations; geometric inequalities; affine geometry; and curvature measures. The book will be of interest to a broad readership, from those involved in Convex Geometry, to those focusing on Functional Analysis, Harmonic Analysis, Differential Geometry, or PDEs. The book is a addressed to PhD students and researchers, interested in Convex Geometry and its links to analysis.

Convex domains. --- Mathematics. --- Algebra. --- Field theory (Physics). --- Group theory. --- Mathematical analysis. --- Analysis (Mathematics). --- Convex geometry. --- Discrete geometry. --- Convex and Discrete Geometry. --- Analysis. --- Field Theory and Polynomials. --- Group Theory and Generalizations. --- Geometry --- Combinatorial geometry --- 517.1 Mathematical analysis --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Math --- Science --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Discrete groups. --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry . --- Algebraic fields. --- Polynomials. --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra)

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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.

Convex geometry. --- Discrete geometry. --- Convex and Discrete Geometry. --- Geometria discreta