V. Milman. page 647--667. Birkhäuser Basel, Basel, (2008)
It was recently observed that asymptotic theory of high-dimensional convexity is extended in a very broad sense to the category of log-concave measures, and moreover, this extension is needed to understand and to solve some problems of asymptotic theory of high-dimensional convexity proper. Many important geometric inequalities were interpreted and extended to such category. On the other hand, some typical probabilisitic results are interpreted and proved in a geometric framework. Even more importantly, such extension to the log-concave category was needed to solve some central problems of a purely geometric nature. The goal of this article is to outline this development and to demonstrate examples of results which confirm this picture.