Flag representations of mixed volumes and mixed functionals of convex bodies

Mixed volumes V(K1,…,Kd) of convex bodies K1,…,Kd in Euclidean space Rd are of central importance in the Brunn-Minkowski theory. Representations for mixed volumes are available in special cases, for example as integrals over the unit sphere with respect to mixed area measures. More generally, in Hug-Rataj-Weil (2013) [11] a formula for V(K[n],M[d−n]), n∈{1,…,d−1}, as a double integral over flag manifolds was established which involved certain flag measures of the convex bodies K and M (and required a general position of the bodies). In the following, we discuss the general case V(K1[n1],…,Kk[nk]), n1+⋯+nk=d, and show a corresponding result involving the flag measures Ωn1(K1;⋅),…,Ωnk(Kk;⋅). For this purpose, we first establish a curvature representation of mixed volumes over the normal bundles of the bodies involved. We also obtain a corresponding flag representation for the mixed functionals from translative integral geometry and a local version, for mixed (translative) curvature measures.