On unbalanced difference bodies and Godbersen's conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915052611698688 |
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| author | Artstein-Avidan, Shiri Putterman, Eli |
| author_facet | Artstein-Avidan, Shiri Putterman, Eli |
| contents | The longstanding Godbersen's conjecture states that for any convex body $K \subset \mathbb R^n$ of volume $1$ and any $j \in \{0, \ldots, n\}$, the mixed volume $V_j = V(K[j], -K[n - j])$ is bounded by $\binom{n}{j}$, with equality if and only if $K$ is a simplex. We demonstrate that several consequences of this conjecture are true: certain families of linear combinations of the $V_j$, arising from different geometric constructions, are bounded above by their values when one substitutes $\binom{n}{j}$ for $V_j$, with equality if and only if $K$ is a simplex. One of our results implies that for any $K$ of volume $1$ we have $\frac{1}{n + 1} \sum_{j = 0}^n \binom{n}{j}^{-1} V_j \le 1$, showing that Godbersen's conjecture holds ``on average'' for any body. Another result generalizes the well-known Rogers-Shephard inequality for the difference body. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On unbalanced difference bodies and Godbersen's conjecture Artstein-Avidan, Shiri Putterman, Eli Metric Geometry 52A39 The longstanding Godbersen's conjecture states that for any convex body $K \subset \mathbb R^n$ of volume $1$ and any $j \in \{0, \ldots, n\}$, the mixed volume $V_j = V(K[j], -K[n - j])$ is bounded by $\binom{n}{j}$, with equality if and only if $K$ is a simplex. We demonstrate that several consequences of this conjecture are true: certain families of linear combinations of the $V_j$, arising from different geometric constructions, are bounded above by their values when one substitutes $\binom{n}{j}$ for $V_j$, with equality if and only if $K$ is a simplex. One of our results implies that for any $K$ of volume $1$ we have $\frac{1}{n + 1} \sum_{j = 0}^n \binom{n}{j}^{-1} V_j \le 1$, showing that Godbersen's conjecture holds ``on average'' for any body. Another result generalizes the well-known Rogers-Shephard inequality for the difference body. |
| title | On unbalanced difference bodies and Godbersen's conjecture |
| topic | Metric Geometry 52A39 |
| url | https://arxiv.org/abs/2412.05308 |