Geometrization of Probability

The framework of the subject we will discuss in this survey involves very high dimensional spaces (normed spaces, convex bodies) and accompanying asymptotic (by increasing dimension) phenomena. The starting point of this direction was the open problems of Geometric Functional Analysis (in the ’60s and ’70s). This development naturally led to the Asymptotic Theory of Finite Dimensional spaces (in ’80s and ’90s). See the books [MS86], [Pi89] and the survey [LM93] where this point of view still prevails. During this period, the problems and methods of Classical Convexity were absorbed by the Asymptotic Theory (including geometric inequalites and many geometric, i.e. “isometric” as opposed to “isomorphic” problems). As an outcome, we derived a new theory: Asymptotic Geometric Analysis. (Two surveys, [GM01] and [GM04] give a proper picture of this theory at this stage.) One of the most important points of already the first stage of this development is a change in intuition about the behavior of high-dimensional spaces. Instead of the diversity expected in high dimensions and chaotic behavior, we observe a unified behavior with very little diversity. We analyze this change of intuition in [M98] and [M00]. We refer the reader to [M00] for some examples which illustrate this. Also in [M04], we attempt to describe the main principles and phenonema governing the asymptotic behavior of high-dimensional convex bodies and normed spaces.