  Flag measures are descriptors of convex bodies $K$ in $d$-dimensional
Euclidean space generalizing the classical area measures. They have been used
to provide general integral formulas for mixed volumes (see Hug, Rataj and Weil
(2017)). Here, we consider an image measure $\gamma_j(K,\cdot)$ of flag
measures, defined on the Grassmannian $G(d,j)$ of affine $j$-spaces, $1\le j\le
d-1$, and show that it determines centrally symmetric bodies $K$ of dimension
$\geq j+1$ uniquely. We then explain that Grassmann measures appear in the
representation of smooth, translation invariant, continuous and even valuations
due to Alesker (2003). Using this connection, we prove a uniqueness result for
projection averages of area measures and we finally discuss a Grassmann version
of the natural touching measure of convex bodies.
