A product integral representation of mixed volumes of two convex bodies

Abstract

The Brunn-Minkowski theory in convex geometry relies heavily on the notion of mixed volumes. Despite its particular importance, even explicit representations for the mixed volumes of two convex bodies in Euclidean space R-d are available only in special cases.

Here we investigate a new integral representation of such mixed volumes, in terms of flag measures of the involved convex sets. A brief introduction to (extended) flag measures of convex bodies is also provided.

Keywords

• Mixed volumes
• flag measures
• Grassmannian
• integral geometry
• generalized curvatures