Strange shadows of $\ell_p$-balls
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916539962228736 |
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| author | Kabluchko, Zakhar Sonnleitner, Mathias |
| author_facet | Kabluchko, Zakhar Sonnleitner, Mathias |
| contents | We prove a large deviations principle for orthogonal projections of the unit ball $\mathbb{B}_p^n$ of $\ell_p^n$ onto a random $k$-dimensional linear subspace of $\mathbb{R}^n$ as $n\to\infty$ in the case $2<p\le \infty$ and for the intersection of $\mathbb{B}_p^n$ with a random $k$-dimensional subspace in the case $1\le p <2$. The corresponding rate function is finite only on $L_q$-zonoids and their duals, respectively, and given in terms of the maximum entropy over suitable measures generating the $L_q$-zonoid, where $\frac{1}{p}+\frac{1}{q}=1$. In particular, we obtain that the renormalized projections/sections almost surely tend to a $k$-dimensional Euclidean ball of certain radius. Moreover, we identify the asymptotic probability that the random orthogonal projection remains within a ball of smaller radius. As a byproduct we obtain an interesting inequality for the Gamma function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strange shadows of $\ell_p$-balls Kabluchko, Zakhar Sonnleitner, Mathias Probability Functional Analysis 52A23 (Primary) 52A21, 52A22, 60F10 (Secondary) We prove a large deviations principle for orthogonal projections of the unit ball $\mathbb{B}_p^n$ of $\ell_p^n$ onto a random $k$-dimensional linear subspace of $\mathbb{R}^n$ as $n\to\infty$ in the case $2<p\le \infty$ and for the intersection of $\mathbb{B}_p^n$ with a random $k$-dimensional subspace in the case $1\le p <2$. The corresponding rate function is finite only on $L_q$-zonoids and their duals, respectively, and given in terms of the maximum entropy over suitable measures generating the $L_q$-zonoid, where $\frac{1}{p}+\frac{1}{q}=1$. In particular, we obtain that the renormalized projections/sections almost surely tend to a $k$-dimensional Euclidean ball of certain radius. Moreover, we identify the asymptotic probability that the random orthogonal projection remains within a ball of smaller radius. As a byproduct we obtain an interesting inequality for the Gamma function. |
| title | Strange shadows of $\ell_p$-balls |
| topic | Probability Functional Analysis 52A23 (Primary) 52A21, 52A22, 60F10 (Secondary) |
| url | https://arxiv.org/abs/2412.17475 |