Log-concavity of mean section operators

Abstract

The Brunn-Minkowski inequality is one of the most central results in modern convex geometry and is closely related to many other important geometric inequalities. In recent years, efforts have been made to generalize the Brunn-Minkowski inequality to compositions of mixed volumes and homogeneous Minkowski valuations. In this thesis, we give an exposition of the generalizations that have been achieved so far, and we examine the open cases. Concrete natural examples of Minkowki valuations for which the generalized Brunn-Minkowki inequality remains open are provided by the so-called mean section operators. In this thesis, we expound a recent proof by P. Goodey and W. Weil of an integral representation formula for the mean section operators. Lastly, we briefly discuss how the Brunn-Minkowski inequality could be extended to larger classes of Minkowski valuations including mean section operators.