Distances between non-symmetric convex bodies: optimal bounds up to polylog
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914138608893952 |
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| author | Bizeul, Pierre Klartag, Boaz |
| author_facet | Bizeul, Pierre Klartag, Boaz |
| contents | We show that the non-symmetric Banach-Mazur distance between two convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$ satisfies $$ d_{BM}(K_1, K_2) \leq C n \cdot \log^α (n+1), $$ for universal constants $C, α> 0$. This improves upon the earlier bound $C n^{4/3} \log^α (n+1)$ due to Rudelson. Up to polylogarithmic factors, our estimate is optimal and it also matches the optimal bound in the centrally-symmetric case which is realized in the John position, as proven by Gluskin. The bound above for the Banach-Mazur distance is attained when both bodies are in a ``random isotropic position'', that is, in isotropic position after a random rotation. Our proof is based on an $M$-bound in the isotropic position, which complements E. Milman's $M^*$-bound. In addition, we consider the partial containment distance $d_{PC}(K_1, K_2)$ between two convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$, where the Banach-Mazur requirement to contain $100\%$ of the other body is relaxed to $99\%$-containment. We prove that for any pair of convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$, $$ d_{PC}(K_1, K_2) \leq C \log^α (n+1), $$ and that any isotropic position of $K_1$ and $K_2$ yields this polylogarithmic bound for $d_{PC}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distances between non-symmetric convex bodies: optimal bounds up to polylog Bizeul, Pierre Klartag, Boaz Metric Geometry Functional Analysis Probability We show that the non-symmetric Banach-Mazur distance between two convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$ satisfies $$ d_{BM}(K_1, K_2) \leq C n \cdot \log^α (n+1), $$ for universal constants $C, α> 0$. This improves upon the earlier bound $C n^{4/3} \log^α (n+1)$ due to Rudelson. Up to polylogarithmic factors, our estimate is optimal and it also matches the optimal bound in the centrally-symmetric case which is realized in the John position, as proven by Gluskin. The bound above for the Banach-Mazur distance is attained when both bodies are in a ``random isotropic position'', that is, in isotropic position after a random rotation. Our proof is based on an $M$-bound in the isotropic position, which complements E. Milman's $M^*$-bound. In addition, we consider the partial containment distance $d_{PC}(K_1, K_2)$ between two convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$, where the Banach-Mazur requirement to contain $100\%$ of the other body is relaxed to $99\%$-containment. We prove that for any pair of convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$, $$ d_{PC}(K_1, K_2) \leq C \log^α (n+1), $$ and that any isotropic position of $K_1$ and $K_2$ yields this polylogarithmic bound for $d_{PC}$. |
| title | Distances between non-symmetric convex bodies: optimal bounds up to polylog |
| topic | Metric Geometry Functional Analysis Probability |
| url | https://arxiv.org/abs/2510.20511 |