Equivariant Valuations on Convex Functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913427031588864 |
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| author | Hofstätter, Georg C. Knoerr, Jonas |
| author_facet | Hofstätter, Georg C. Knoerr, Jonas |
| contents | We classify all continuous valuations on the space of finite convex functions with values in the same space which are dually epi-translation-invariant and equi- resp. contravariant with respect to volume-preserving linear maps. We thereby identify the valuation-theoretic functional analogues of the difference body map and show that there does not exist a generalization of the projection body map in this setting. This non-existence result is shown to also hold true for valuations with values in the space of convex functions that are finite in a neighborhood of the origin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08304 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Valuations on Convex Functions Hofstätter, Georg C. Knoerr, Jonas Metric Geometry Functional Analysis 52A41, 26B25, 52B45 We classify all continuous valuations on the space of finite convex functions with values in the same space which are dually epi-translation-invariant and equi- resp. contravariant with respect to volume-preserving linear maps. We thereby identify the valuation-theoretic functional analogues of the difference body map and show that there does not exist a generalization of the projection body map in this setting. This non-existence result is shown to also hold true for valuations with values in the space of convex functions that are finite in a neighborhood of the origin. |
| title | Equivariant Valuations on Convex Functions |
| topic | Metric Geometry Functional Analysis 52A41, 26B25, 52B45 |
| url | https://arxiv.org/abs/2407.08304 |