Additive kinematic formulas for convex functions

Abstract

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge-Ampère measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge-Ampère measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\mathrm{SO}(n-1)\times O(1)$ is considered.