Stability result for the extremal Grünbaum distance between convex bodies
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866911551553798144 |
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| author | Kobos, Tomasz |
| author_facet | Kobos, Tomasz |
| contents | In 1963 Grünbaum introduced the following variation of the Banach-Mazur distance for arbitrary convex bodies $K, L \subset \mathbb{R}^n$: $d_G(K, L) = \inf \{ |r| \ : \ K' \subset L' \subset rK' \}$ with the infimum taken over all non-degenerate affine images $K'$ and $L'$ of $K$ and $L$ respectively. In 2004 Gordon, Litvak, Meyer and Pajor proved that the maximal possible distance is equal to $n$, confirming the conjecture of Grünbaum. In 2011 Jiménez and Naszódi asked if the equality $d_G(K, L)=n$ implies that $K$ or $L$ is a simplex and they proved it under the additional assumption that one of the bodies is smooth or strictly convex. The aim of the paper is to give a stability result for a smooth case of the theorem of Jiménez and Naszódi. We prove that for each smooth convex body $L$ there exists $\varepsilon_0(L)>0$ such that if $d_G(K, L) \geq (1-\varepsilon)n$ for some $0 \leq \varepsilon \leq \varepsilon_0(L)$, then $d(K, S_n) \leq 1 + 40n^3r(\varepsilon)$, where $S_n$ is the simplex in $\mathbb{R}^n$, $r(\varepsilon)$ is a specific function of $\varepsilon$ depending on the modulus of the convexity of the polar body of $L$ and $d$ is the usual Banach-Mazur distance. As a consequence, we obtain that for arbitrary convex bodies $K, L \subset \mathbb{R}^n$ their Banach-Mazur distance is less than $n^2 - 2^{-22}n^{-7}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1710_01170 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Stability result for the extremal Grünbaum distance between convex bodies Kobos, Tomasz Functional Analysis Primary 52A40, 52A20, 52A27 In 1963 Grünbaum introduced the following variation of the Banach-Mazur distance for arbitrary convex bodies $K, L \subset \mathbb{R}^n$: $d_G(K, L) = \inf \{ |r| \ : \ K' \subset L' \subset rK' \}$ with the infimum taken over all non-degenerate affine images $K'$ and $L'$ of $K$ and $L$ respectively. In 2004 Gordon, Litvak, Meyer and Pajor proved that the maximal possible distance is equal to $n$, confirming the conjecture of Grünbaum. In 2011 Jiménez and Naszódi asked if the equality $d_G(K, L)=n$ implies that $K$ or $L$ is a simplex and they proved it under the additional assumption that one of the bodies is smooth or strictly convex. The aim of the paper is to give a stability result for a smooth case of the theorem of Jiménez and Naszódi. We prove that for each smooth convex body $L$ there exists $\varepsilon_0(L)>0$ such that if $d_G(K, L) \geq (1-\varepsilon)n$ for some $0 \leq \varepsilon \leq \varepsilon_0(L)$, then $d(K, S_n) \leq 1 + 40n^3r(\varepsilon)$, where $S_n$ is the simplex in $\mathbb{R}^n$, $r(\varepsilon)$ is a specific function of $\varepsilon$ depending on the modulus of the convexity of the polar body of $L$ and $d$ is the usual Banach-Mazur distance. As a consequence, we obtain that for arbitrary convex bodies $K, L \subset \mathbb{R}^n$ their Banach-Mazur distance is less than $n^2 - 2^{-22}n^{-7}$. |
| title | Stability result for the extremal Grünbaum distance between convex bodies |
| topic | Functional Analysis Primary 52A40, 52A20, 52A27 |
| url | https://arxiv.org/abs/1710.01170 |