  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
$\Omega_{n_1}(K_1;\cdot),\dots, \Omega_{n_k}(K_k;\cdot)$. 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.
