Weighted surface area measures and Brunn-Minkowski theory

The Brunn-Minkowski theory in convex geometry concerns, among other things, the study of volumes, mixed volumes, and surface area measures of convex bodies. In this talk, we review the generalizations of these concepts to Borel measures with density in $n$-dimensional real Euclidean space, such as weighted surface area and mixed measures. We then introduce the mixed measures of three bodies and explore inequalities related to this quantity.