On a generalization of Godbersen's conjecture
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914120043855872 |
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| author | Kotrbatý, Jan |
| author_facet | Kotrbatý, Jan |
| contents | The long-standing Godbersen's conjecture asserts that the Rogers-Shephard inequality for the volume of the difference body is refined by an inequality for the mixed volume of a convex body and its reflection about the origin. The conjecture is known in several special cases, notably for anti-blocking convex bodies. In this note, we propose a generalization of Godbersen's conjecture that refines Schneider's generalization of the Rogers-Shephard inequality to higher-order difference bodies and prove our conjecture for anti-blocking convex bodies. Moreover, we relate the conjectured inequality to the higher-rank mixed volume defined by the author and Wannerer which leads to an equivalent formulation in terms of the Alesker product of smooth, translation invariant valuations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a generalization of Godbersen's conjecture Kotrbatý, Jan Metric Geometry 52A40, 52A39, 52B45, 52B12 The long-standing Godbersen's conjecture asserts that the Rogers-Shephard inequality for the volume of the difference body is refined by an inequality for the mixed volume of a convex body and its reflection about the origin. The conjecture is known in several special cases, notably for anti-blocking convex bodies. In this note, we propose a generalization of Godbersen's conjecture that refines Schneider's generalization of the Rogers-Shephard inequality to higher-order difference bodies and prove our conjecture for anti-blocking convex bodies. Moreover, we relate the conjectured inequality to the higher-rank mixed volume defined by the author and Wannerer which leads to an equivalent formulation in terms of the Alesker product of smooth, translation invariant valuations. |
| title | On a generalization of Godbersen's conjecture |
| topic | Metric Geometry 52A40, 52A39, 52B45, 52B12 |
| url | https://arxiv.org/abs/2502.02149 |