David Alonso-Gutiérrez – författare

These proceedings result from the International Conference 'Geometry, Analysis & Convexity' (OLE 2022) held from 20th to 24th June 2022 at the Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Spain and they include some of the contributions presented at this conference. This book is addressed to any researcher interested in convex geometric analysis and asymptotic analysis as well as integral geometry and discrete geometry and their applications in convexity, and related topics. Convex geometric analysis was born from the increasing interaction between classical (convex) geometry and asymptotic (convex) analysis. During the last three decades, the study of the integral geometry of convex bodies has been fuelled by the introduction of methods, results and new points of view coming from other branches of mathematics such as probability, harmonic analysis, geometry of finite dimensional normed spaces, integral geometry and discrete geometry. These recent advances have revealed fruitful connections between geometric inequalities, transport theory and information theory. Asymptotic convex analysis is mainly concerned with geometric properties of convex bodies in finite dimensional normed spaces, focused when the dimension tends to infinity. The understanding of high dimensional phenomena becomes an important point since high dimensional problems are frequently encountered in mathematics and applied sciences. Concentration of measure phenomenon can be viewed as an isoperimetric problem, which lies at the heart of classical geometry and calculus of variation. Besides convex geometry, geometric analysis has been developed using techniques and deep theorems from integral geometry, where the notion of measure is generalized to the concept of the so-called valuation, and it has developed from a simple technique to a fundamental area, the theory of valuations. The underlying structure of the valuation space (invariant under translations) is intrinsically connected with affine or analytic isoperimetric inequalities, among others. It is addressed to researchers in this field.

These proceedings result from the International Conference 'Geometry, Analysis & Convexity' (OLE 2022) held from 20th to 24th June 2022 at the Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Spain and they include some of the contributions presented at this conference. This book is addressed to any researcher interested in convex geometric analysis and asymptotic analysis as well as integral geometry and discrete geometry and their applications in convexity, and related topics. Convex geometric analysis was born from the increasing interaction between classical (convex) geometry and asymptotic (convex) analysis. During the last three decades, the study of the integral geometry of convex bodies has been fuelled by the introduction of methods, results and new points of view coming from other branches of mathematics such as probability, harmonic analysis, geometry of finite dimensional normed spaces, integral geometry and discrete geometry. These recent advances have revealed fruitful connections between geometric inequalities, transport theory and information theory. Asymptotic convex analysis is mainly concerned with geometric properties of convex bodies in finite dimensional normed spaces, focused when the dimension tends to infinity. The understanding of high dimensional phenomena becomes an important point since high dimensional problems are frequently encountered in mathematics and applied sciences. Concentration of measure phenomenon can be viewed as an isoperimetric problem, which lies at the heart of classical geometry and calculus of variation. Besides convex geometry, geometric analysis has been developed using techniques and deep theorems from integral geometry, where the notion of measure is generalized to the concept of the so-called valuation, and it has developed from a simple technique to a fundamental area, the theory of valuations. The underlying structure of the valuation space (invariant under translations) is intrinsically connected with affine or analytic isoperimetric inequalities, among others. It is addressed to researchers in this field.

Focusing on two central conjectures of Asymptotic Geometric Analysis, the Kannan-Lovász-Simonovits spectral gap conjecture and the variance conjecture, these Lecture Notes present the theory in an accessible way, so that interested readers, even those who are not experts in the field, will be able to appreciate the treated topics. Offering a presentation suitable for professionals with little background in analysis, geometry or probability, the work goes directly to the connection between isoperimetric-type inequalities and functional inequalities, giving the interested reader rapid access to the core of these conjectures.In addition, four recent and important results in this theory are presented in a compelling way. The first two are theorems due to Eldan-Klartag and Ball-Nguyen, relating the variance and the KLS conjectures, respectively, to the hyperplane conjecture. Next, the main ideas needed prove the best known estimate for the thin-shell width given by Guédon-Milman and an approach to Eldan's work on the connection between the thin-shell width and the KLS conjecture are detailed.