Translation invariant area measures on convex bodies
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913172259078144 |
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| author | Knoerr, Jonas |
| author_facet | Knoerr, Jonas |
| contents | We introduce the space of continuous and translation invariant area measures, which are measure-valued functionals on the space of convex bodies satisfying a certain locality condition. Our main result shows that the space of $\mathrm{GL}(n,\mathbb{R})$-smooth area measures coincides with the space of area measures obtained by integration with respect to the normal cycle. We show how this result yields Hadwiger-type classification results for continuous area measures that are equivariant with respect to compact groups acting transitively on the unit sphere. In addition, we establish a general density criterion for invariant submodules and show that mixed area measures generate dense submodules with respect to suitable topologies on the space of continuous area measures. As a byproduct, we discuss how McMullen's Conjecture can be obtained directly from the representation of $\mathrm{GL}(n,\mathbb{R})$-smooth translation invariant valuations on convex bodies in terms of integration with respect to the normal cycle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30927 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Translation invariant area measures on convex bodies Knoerr, Jonas Metric Geometry 52B45, 52A20, 53C65, 52A39 We introduce the space of continuous and translation invariant area measures, which are measure-valued functionals on the space of convex bodies satisfying a certain locality condition. Our main result shows that the space of $\mathrm{GL}(n,\mathbb{R})$-smooth area measures coincides with the space of area measures obtained by integration with respect to the normal cycle. We show how this result yields Hadwiger-type classification results for continuous area measures that are equivariant with respect to compact groups acting transitively on the unit sphere. In addition, we establish a general density criterion for invariant submodules and show that mixed area measures generate dense submodules with respect to suitable topologies on the space of continuous area measures. As a byproduct, we discuss how McMullen's Conjecture can be obtained directly from the representation of $\mathrm{GL}(n,\mathbb{R})$-smooth translation invariant valuations on convex bodies in terms of integration with respect to the normal cycle. |
| title | Translation invariant area measures on convex bodies |
| topic | Metric Geometry 52B45, 52A20, 53C65, 52A39 |
| url | https://arxiv.org/abs/2605.30927 |