In this book, we present a collection of results situated at the confluence of two vibrant and rapidly developing areas within convex geometry: the functional approach, which seeks to reinterpret and extend classical theorems of convex geometry by replacing volume with more general measures, and geometric tomography, a field concerned with recovering and analyzing geometric properties of convex bodies through partial data, such as the measures of their sections or projections. We refer to the synthesis of these two perspectives as the tomography of measures.

Among the central problems discussed in the book are the functional analogue of the classical Busemann–Petty problem and the slicing inequality for measures. These problems lie at the heart of asymptotic geometric analysis and reflect ongoing efforts to understand dimension-dependent phenomena in high-dimensional spaces.

The book is intended for mathematicians with an interest in geometry, harmonic analysis, functional analysis, and probability theory.