We give a geometric description of the support of a special class of mixed area measures of convex bodies. We thereby confirm a particular case of a conjecture by Schneider. Our results are motivated by problems on functional intrinsic volumes and mixed Monge-Ampère measures. We show how answers to these analytic questions follow from the aforementioned results on mixed area measures of convex bodies. We also obtain new integral geometric formulas for both convex bodies and convex functions. Joint work with Daniel Hug and Jacopo Ulivelli.