Abstrakt
Mixed volumes V(K-1, ... , K-d) of convex bodies K-1, ... , K-d in Euclidean space R-d 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 is an element of {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(K-1[n(1)], ... , K-k[n(k)]), n(1) + ... + n(k) = d, and show a corresponding result involving the flag measures Omega(n1) (K-1;.), ... , Omega(nk) (K-k;.).
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.