Convexity theory studies geometry of convex bodies in Euclidean space of a fixed dimension. It is relevant in numerous mathematical disciplines: geometric tomography, stochastic and integral geometry, non-linear PDE's, Fourier analysis, functional analysis, mathematical programming and optimization.
High dimensional phenomena appear naturally in a number of areas of mathematics and adjacent domains of science, particularly in computer science and physics, dealing with objects that are determined by an infinitely growing number of parameters and modeled by functions of an infinitely growing number of variables. They can be analyzed using deep analytic, geometric, probabilistic and combinatorial tools, many of which originally arose in geometric functional analysis (a.k.a. local theory of Banach spaces) or, in an independent development, in theoretical computer science. The results in this area have applications not only in other areas of mathematics such as analysis, geometry and probability theory, but also in operations research, control theory, computer science, statistics and mathematical or statistical physics.
Convexity theory and the theory of high dimensional phenomena are now closely interconnected. Developments in one area have enriched the other, and links to other fields are being discovered continuously.