# Convex geometry and its connections to harmonic analysis, functional analysis and probability theory

### Keith Ball

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

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This book chapter is published *open access.*

## Abstract

Convex geometry and analysis have connections to many areas of the mathematical sciences: PDEs, discrete geometry, optimization, theoretical computer science, and mathematical economics. No article could even scratch the surface of all of these. Instead, we shall begin by describing how the development of the subject was influenced over the last 50 years by two other fields, harmonic and functional analysis, and then discuss the subtle and still somewhat mysterious way in which convex domains exhibit properties that we normally expect to see within probability theory.