Theoretical study demonstrates a functional additive kinematic formula for convex functions, highlighting unified representations of functional intrinsic volumes.
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-Amp\`ere 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-Amp\`ere measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on SO(n-1)× O(1) is considered.
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Hug et al. (2024) studied this question.
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