On hyperplane sections and projections in $l_p^n$
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| Format: | Preprint |
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2024
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| _version_ | 1866913276807348224 |
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| author | König, Hermann |
| author_facet | König, Hermann |
| contents | For $2 < p < p_0 \simeq 26.265$, the hyperplane section of the $l_p^n$-unit ball $B_p^n$ perpendicular to a^(n) = 1/sqrt(n) (1, ... ,1) for large $n$ has larger volume than the one orthogonal to a^(2) = 1/sqrt(2) (1,1,0, ...,0), as shown by Oleszkiewicz. This is different from the case of $l_\infty^n$ considered by Ball. We give a quantitative estimate for which dimensions $n$ this happens, namely for $n > c (\frac 1 {p_0-p} + \frac 1 {p-2})$ for some absolute constant $c>0$. Correspondingly for projections of $B_q^n$ onto hyperplanes, Barthe and Naor showed that projections onto hyperplanes perpendicular to $a^{(n)}$ have smaller volume for large $n$ than onto the one orthogonal to $a^{(2)}$, if $\frac 4 3 < q < 2$, different from the case $q=1$. We show that this happens for all $n > 5 (\frac 1 {q-\frac 4 3} + \frac 1 {2-q})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_14456 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On hyperplane sections and projections in $l_p^n$ König, Hermann Functional Analysis 52A38, 46B07 For $2 < p < p_0 \simeq 26.265$, the hyperplane section of the $l_p^n$-unit ball $B_p^n$ perpendicular to a^(n) = 1/sqrt(n) (1, ... ,1) for large $n$ has larger volume than the one orthogonal to a^(2) = 1/sqrt(2) (1,1,0, ...,0), as shown by Oleszkiewicz. This is different from the case of $l_\infty^n$ considered by Ball. We give a quantitative estimate for which dimensions $n$ this happens, namely for $n > c (\frac 1 {p_0-p} + \frac 1 {p-2})$ for some absolute constant $c>0$. Correspondingly for projections of $B_q^n$ onto hyperplanes, Barthe and Naor showed that projections onto hyperplanes perpendicular to $a^{(n)}$ have smaller volume for large $n$ than onto the one orthogonal to $a^{(2)}$, if $\frac 4 3 < q < 2$, different from the case $q=1$. We show that this happens for all $n > 5 (\frac 1 {q-\frac 4 3} + \frac 1 {2-q})$. |
| title | On hyperplane sections and projections in $l_p^n$ |
| topic | Functional Analysis 52A38, 46B07 |
| url | https://arxiv.org/abs/2403.14456 |