  Motivated by a problem for mixed Monge-Amp\`ere measures of convex functions,
we address a special case of a conjecture of Schneider and show that for every
convex body $K$ the support of the mixed area measure
$S(K[j],B_L^{n-1}[n-1-j],\cdot)$ is given by the set of
$(K[j],B_L^{n-1}[n-1-j])$-extreme unit normal vectors, where $B_L^{n-1}$
denotes the $(n-1)$-dimensional Euclidean unit ball in a hyperplane $L$. As a
consequence, we see that the supports of these measures are nested. Our proof
introduces a Kubota-type formula for mixed area measures, which involves
integration over $j$-dimensional linear subspaces that contain a fixed
$1$-dimensional subspace. We transfer these results to the analytic setting,
where we obtain corresponding statements for (conjugate) mixed Monge-Amp\`ere
measures of convex functions. Thereby, we establish a fundamental property for
functional intrinsic volumes. In addition, we study connections between mixed
Monge-Amp\`ere measures of convex functions and mixed area measures and mixed
volumes of convex bodies.
