  The $k$th projection function $v_k(K,\cdot)$ of a convex body $K\subset
{\mathbb R}^d, d\ge 3,$ is a function on the Grassmannian $G(d,k)$ which
measures the $k$-dimensional volume of the projection of $K$ onto members of
$G(d,k)$. For $k=1$ and $k=d-1$, simple formulas for the projection functions
exist. In particular, $v_{d-1}(K,\cdot)$ can be written as a spherical integral
with respect to the surface area measure of $K$. Here, we generalize this
result and prove two integral representations for $v_k(K,\cdot),
k=1,\dots,d-1$, over flag manifolds. Whereas the first representation
generalizes a result of Ambartzumian (1987), but uses a flag measure which is
not continuous in $K$, the second representation is related to a recent flag
formula for mixed volumes by Hug, Rataj and Weil (2013) and depends
continuously on $K$.
