Kinematic formulas in convex geometry for non-compact groups
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913785953910784 |
|---|---|
| author | Anjos, Sílvia Nascimento, Francisco |
| author_facet | Anjos, Sílvia Nascimento, Francisco |
| contents | We generalize classical kinematic formulas for convex bodies in a real vector space $V$ to the setting of non-compact Lie groups admitting a Cartan decomposition. Specifically, let $G$ be a closed linear group with Cartan decomposition $G \cong K \times \exp(\mathfrak{p}_0)$, where $K$ is a maximal compact subgroup acting transitively on the unit sphere. For $K$-invariant continuous valuations on convex bodies, we establish an integral geometric-type formula for $\overline{G} = G \ltimes V$. Key to our approach is the introduction of a Gaussian measure on $\mathfrak{p}_0$, which ensures convergence of the non-compact part of the integral. In the special case $K = O(n)$, we recover a Hadwiger-type formula involving intrinsic volumes, with explicit constants $c_j$ computed via a Weyl integration formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kinematic formulas in convex geometry for non-compact groups Anjos, Sílvia Nascimento, Francisco Metric Geometry Primary 53C65, Secondary 52A22, 22E46 We generalize classical kinematic formulas for convex bodies in a real vector space $V$ to the setting of non-compact Lie groups admitting a Cartan decomposition. Specifically, let $G$ be a closed linear group with Cartan decomposition $G \cong K \times \exp(\mathfrak{p}_0)$, where $K$ is a maximal compact subgroup acting transitively on the unit sphere. For $K$-invariant continuous valuations on convex bodies, we establish an integral geometric-type formula for $\overline{G} = G \ltimes V$. Key to our approach is the introduction of a Gaussian measure on $\mathfrak{p}_0$, which ensures convergence of the non-compact part of the integral. In the special case $K = O(n)$, we recover a Hadwiger-type formula involving intrinsic volumes, with explicit constants $c_j$ computed via a Weyl integration formula. |
| title | Kinematic formulas in convex geometry for non-compact groups |
| topic | Metric Geometry Primary 53C65, Secondary 52A22, 22E46 |
| url | https://arxiv.org/abs/2504.06743 |