Mahler-type volume inequality for convex bodies with tetrahedral symmetry
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914164063076352 |
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| author | Aliev, Arkadiy |
| author_facet | Aliev, Arkadiy |
| contents | Let $ K $ be a convex body in $ \mathbb{R}^n $. We denote the volume of $ K $ by $ \vert K\vert $, and the polar body of its difference body $ K - K $ by $ (K - K)^{\circ} $. We provide a new proof of the well-known estimate \[ |K||(K - K)^{\circ}| \geq \frac{3}{2} \] for $ K \subset \mathbb{R}^2 $, with equality attained for a triangle. For $ K \subset \mathbb{R}^3 $ with tetrahedral symmetry, we prove that \[ |K| |(K - K)^{\circ}| \geq \frac{2}{3}, \] with equality attained for a tetrahedron. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mahler-type volume inequality for convex bodies with tetrahedral symmetry Aliev, Arkadiy Metric Geometry Let $ K $ be a convex body in $ \mathbb{R}^n $. We denote the volume of $ K $ by $ \vert K\vert $, and the polar body of its difference body $ K - K $ by $ (K - K)^{\circ} $. We provide a new proof of the well-known estimate \[ |K||(K - K)^{\circ}| \geq \frac{3}{2} \] for $ K \subset \mathbb{R}^2 $, with equality attained for a triangle. For $ K \subset \mathbb{R}^3 $ with tetrahedral symmetry, we prove that \[ |K| |(K - K)^{\circ}| \geq \frac{2}{3}, \] with equality attained for a tetrahedron. |
| title | Mahler-type volume inequality for convex bodies with tetrahedral symmetry |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2511.14991 |