On the polar of Schneider's difference body
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916945416159232 |
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| author | Haddad, Julián Langharst, Dylan Livshyts, Galyna V. Putterman, Eli |
| author_facet | Haddad, Julián Langharst, Dylan Livshyts, Galyna V. Putterman, Eli |
| contents | In 1970, Schneider introduced the $m$th-order extension of the difference body $DK$ of a convex body $K\subset\mathbb R^n$, the convex body $D^m(K)$ in $\mathbb R^{nm}$. He conjectured that its volume is minimized for ellipsoids when the volume of $K$ is fixed.
In this work, we solve a dual version of this problem: we show that the volume of the polar body of $D^m(K)$ is maximized precisely by ellipsoids. For $m=1$ this recovers the symmetric case of the celebrated Blaschke-Santaló inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically á la Bourgain-Milman. We also consider a functional version. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the polar of Schneider's difference body Haddad, Julián Langharst, Dylan Livshyts, Galyna V. Putterman, Eli Metric Geometry Functional Analysis 52A40, Secondary: 28A75 In 1970, Schneider introduced the $m$th-order extension of the difference body $DK$ of a convex body $K\subset\mathbb R^n$, the convex body $D^m(K)$ in $\mathbb R^{nm}$. He conjectured that its volume is minimized for ellipsoids when the volume of $K$ is fixed. In this work, we solve a dual version of this problem: we show that the volume of the polar body of $D^m(K)$ is maximized precisely by ellipsoids. For $m=1$ this recovers the symmetric case of the celebrated Blaschke-Santaló inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically á la Bourgain-Milman. We also consider a functional version. |
| title | On the polar of Schneider's difference body |
| topic | Metric Geometry Functional Analysis 52A40, Secondary: 28A75 |
| url | https://arxiv.org/abs/2503.06191 |