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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2412.18444 |
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| _version_ | 1866917205276360704 |
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| author | Ivanov, G. |
| author_facet | Ivanov, G. |
| contents | John's inclusion states that a convex body in $\mathbb{R}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish the following asymptotically tight inequality: \\ \noindent For any log-concave function $f$ with finite, positive integral, there exist a positive definite matrix $A$, a point $a \in \mathbb{R}^d$, and a positive constant $α$ such that \[ χ_{\mathbf{B}^{d}}(x) \leq αf\!\!\left(A(x-a)\right) \leq \sqrt{d+1} \cdot e^{-\frac{\left|x\right|}{d+2} + (d+1)}, \] where $χ_{\mathbf{B}^{d}}$ is the indicator function of the unit ball $\mathbf{B}^{d}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The John inclusion for log-concave functions Ivanov, G. Metric Geometry 52A23 (primary), 52A40, 46T12 John's inclusion states that a convex body in $\mathbb{R}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish the following asymptotically tight inequality: \\ \noindent For any log-concave function $f$ with finite, positive integral, there exist a positive definite matrix $A$, a point $a \in \mathbb{R}^d$, and a positive constant $α$ such that \[ χ_{\mathbf{B}^{d}}(x) \leq αf\!\!\left(A(x-a)\right) \leq \sqrt{d+1} \cdot e^{-\frac{\left|x\right|}{d+2} + (d+1)}, \] where $χ_{\mathbf{B}^{d}}$ is the indicator function of the unit ball $\mathbf{B}^{d}$. |
| title | The John inclusion for log-concave functions |
| topic | Metric Geometry 52A23 (primary), 52A40, 46T12 |
| url | https://arxiv.org/abs/2412.18444 |