Local $L^p$-Brunn-Minkowski Inequalities for $p PDF Download

Author: Alexander V. Kolesnikov
Publisher: American Mathematical Society
ISBN: 1470451603
Category : Mathematics
Languages : en
Pages : 78

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Author: Andrea Colesanti
Publisher:
ISBN:
Category :
Languages : en
Pages : 19

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Author: Yongsheng Han
Publisher: American Mathematical Society
ISBN: 1470453452
Category : Mathematics
Languages : en
Pages : 118

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Author: Ronen Eldan
Publisher: Springer Nature
ISBN: 3031263006
Category : Mathematics
Languages : en
Pages : 443

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This book reflects general trends in the study of geometric aspects of functional analysis, understood in a broad sense. A classical theme in the local theory of Banach spaces is the study of probability measures in high dimension and the concentration of measure phenomenon. Here this phenomenon is approached from different angles, including through analysis on the Hamming cube, and via quantitative estimates in the Central Limit Theorem under thin-shell and related assumptions. Classical convexity theory plays a central role in this volume, as well as the study of geometric inequalities. These inequalities, which are somewhat in spirit of the Brunn-Minkowski inequality, in turn shed light on convexity and on the geometry of Euclidean space. Probability measures with convexity or curvature properties, such as log-concave distributions, occupy an equally central role and arise in the study of Gaussian measures and non-trivial properties of the heat flow in Euclidean spaces. Also discussed are interactions of this circle of ideas with linear programming and sampling algorithms, including the solution of a question in online learning algorithms using a classical convexity construction from the 19th century.

Author: Rolf Schneider
Publisher: Cambridge University Press
ISBN: 1107601010
Category : Mathematics
Languages : en
Pages : 759

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A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.

Author: Tommy Bonnesen
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 192

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Author: Trista A. Mullin
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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The Brunn-Minkowski Inequality is a classical result that compares the volumes of twosets, in particular convex bodies, and the volume of their Minkowski sum. The proof iselegant and the eects are far reaching in mathematics. In this thesis we will examinethe proof of the inequality, and its multiplicative and integral forms. From there wewill explore a few applications and an analog to Brunn's slice theorem. Additionally, wewill look at how the Brunn-Minkowski Inequality can be used to obtain results regardinggeneral log concave measures, isoperimetric inequalities, and spherical concentrations.We will end the journey with a quick look at what can be said about the intersectionbody of a convex body.

Author: Chris Kottke
Publisher: American Mathematical Society
ISBN: 1470455412
Category : Mathematics
Languages : en
Pages : 124

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Author: Shiri Artstein-Avidan
Publisher: Springer Nature
ISBN: 3031378830
Category : Mathematics
Languages : en
Pages : 304

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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.

Author: Alexander Koldobsky
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110775387
Category : Mathematics
Languages : en
Pages : 480

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In recent years, the interaction between harmonic analysis and convex geometry has increased which has resulted in solutions to several long-standing problems. This collection is based on the topics discussed during the Research Semester on Harmonic Analysis and Convexity at the Institute for Computational and Experimental Research in Mathematics in Providence RI in Fall 2022. The volume brings together experts working in related fields to report on the status of major problems in the area including the isomorphic Busemann-Petty and slicing problems for arbitrary measures, extremal problems for Fourier extension and extremal problems for classical singular integrals of martingale type, among others.