Flag representations of mixed volumes and mixed functionals of convex bodies

Mixed volumes $V(K_1,\dots ,K_d)$ of convex bodies $K_1,\dots ,K_d$ in Euclidean space ${\mathbb{R}}^d$ 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) a formula for $V(K[n],M[d-n])$, $n\in \{1,\dots ,d-1\}$, 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 $V(K_1[n_1],\dots ,K_k[n_k])$, $n_1+\cdots +n_k=d$, and show a corresponding result involving the flag measures ${\mathrm{\Omega }}_{n_1}(K_1;\cdot ),\dots ,{\mathrm{\Omega }}_{n_k}(K_k;\cdot )$. 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.