Additive kinematic formulas for convex functions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918367663751168 |
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| author | Hug, Daniel Mussnig, Fabian Ulivelli, Jacopo |
| author_facet | Hug, Daniel Mussnig, Fabian Ulivelli, Jacopo |
| contents | 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 $\operatorname{SO}(n-1)\times \operatorname{O}(1)$ is considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06697 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Additive kinematic formulas for convex functions Hug, Daniel Mussnig, Fabian Ulivelli, Jacopo Metric Geometry Functional Analysis 52A22 (Primary) 26B25, 52A20, 52A39, 52A41, 52B45 (Secondary) 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 $\operatorname{SO}(n-1)\times \operatorname{O}(1)$ is considered. |
| title | Additive kinematic formulas for convex functions |
| topic | Metric Geometry Functional Analysis 52A22 (Primary) 26B25, 52A20, 52A39, 52A41, 52B45 (Secondary) |
| url | https://arxiv.org/abs/2403.06697 |