Translation invariant curvature measures of convex bodies
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918328480563200 |
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| author | Schuhmacher, Jakob Wannerer, Thomas |
| author_facet | Schuhmacher, Jakob Wannerer, Thomas |
| contents | In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In this paper, we continue this line of research by introducing new tools to study curvature measures. Our main results suggest that the space of curvature measures, which is graded by degree and parity, is highly structured: We conjecture that each graded component has length at most $2$ as a representation of the general linear group, and we prove this in degrees $0$ and $n-2$. Beyond this conjectural picture, our methods yield a characterization of Federer's curvature measures under weaker assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14193 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Translation invariant curvature measures of convex bodies Schuhmacher, Jakob Wannerer, Thomas Differential Geometry Metric Geometry In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In this paper, we continue this line of research by introducing new tools to study curvature measures. Our main results suggest that the space of curvature measures, which is graded by degree and parity, is highly structured: We conjecture that each graded component has length at most $2$ as a representation of the general linear group, and we prove this in degrees $0$ and $n-2$. Beyond this conjectural picture, our methods yield a characterization of Federer's curvature measures under weaker assumptions. |
| title | Translation invariant curvature measures of convex bodies |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2601.14193 |