Kubota-type formulas and supports of mixed measures

Kubota's integral formula expresses the intrinsic volumes of a convex body as averages over its projections onto linear subspaces. In this work, we introduce a new class of Kubota-type formulas for mixed area measures adapted to rotations around a fixed axis, which encode a crucial disintegration property. Our construction is motivated by applications to valuations on convex functions. In the latter framework, we obtain corresponding statements for (conjugate) mixed Monge-Amp\`ere measures. As a by-product, we characterize supports of mixed area and mixed Monge-Amp\`ere measures, thereby confirming a special case of a conjecture by Schneider.