Geometry of log-concave functions and measures

We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn-Minkowski and the Blaschke-Santalo inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman's quotient of subspace theorem, and present a functional version of the Urysohn inequality.