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arXiv · 2403.06697

Additive kinematic formulas for convex functions

Daniel Hug·Fabian Mussnig·Jacopo Ulivelli
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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-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 $\operatorname{SO}(n-1)\times \operatorname{O}(1)$ is considered.

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Daniel Hug, Fabian Mussnig, Jacopo Ulivelli. 2024-03-11. Additive kinematic formulas for convex functions. https://doi.org/10.4153/s0008414x24000944

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