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–Santaló 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.Mathematics Subject Classiffications (2000). 52A20, 52A40, 46B07