Weighted Brunn-Minkowski Theory I: On weighted surface area measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed and surface area measures of bodies. We study generalizations these concepts to Borel with density Rn– particular, weighted versions volumes (the so-called measures) when dealing up three distinct then formulate analyze classical measures, obtain a new integral formula for measure As an application, we prove Bézout-type inequality rotational invariant log-concave generalizing result by Artstein-Avidan, Florentin Ostrover. results are interesting even special case standard Gaussian measure.