The lower dimensional slicing inequality for functions and related distance inequalities
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| Format: | Preprint |
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2024
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| _version_ | 1866929578273931264 |
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| author | Haddad, J. |
| author_facet | Haddad, J. |
| contents | It was shown in [11] that for every origin-symmetric star body $K \subseteq \mathbb R^n$ of volume $1$, every even continuous probability density $f$ on $K$ and $1 \leq k \leq n-1$, there exists a subspace $F \subseteq \mathbb R^n$ of codimension $k$ such that
\[
\int_{K \cap F} f \geq c^k (d_{\rm ovr}(K, \mathcal{BP}_k^n))^{-k}
\]
where $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ is the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies, and $c>0$ is a universal constant.
The upper bound $d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}$ was established in [13] for every origin-symmetric convex body $K$.
In this note we show that there exist an origin-symmetric convex body $K$ of volume $1$ and an even continuous probability density $f$ supported on $K$ such that for every subspace $F$ of codimension $k$,
\[
\int_{K \cap F} f \leq \left( c \sqrt{\frac n{k \log(n)} } \right)^{-k}.
\]
As a consequence we obtain a lower bound for $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ with $K$ a convex body, complementing the upper bound in \cite{koldobsky2011isomorphic}. This is \[c \sqrt{n/k} (\log(n))^{-1/2} \leq \sup_K d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}.\]
The case $k=1$ was obtained previously in [5,6]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The lower dimensional slicing inequality for functions and related distance inequalities Haddad, J. Metric Geometry 52A20, 44A12 It was shown in [11] that for every origin-symmetric star body $K \subseteq \mathbb R^n$ of volume $1$, every even continuous probability density $f$ on $K$ and $1 \leq k \leq n-1$, there exists a subspace $F \subseteq \mathbb R^n$ of codimension $k$ such that \[ \int_{K \cap F} f \geq c^k (d_{\rm ovr}(K, \mathcal{BP}_k^n))^{-k} \] where $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ is the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies, and $c>0$ is a universal constant. The upper bound $d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}$ was established in [13] for every origin-symmetric convex body $K$. In this note we show that there exist an origin-symmetric convex body $K$ of volume $1$ and an even continuous probability density $f$ supported on $K$ such that for every subspace $F$ of codimension $k$, \[ \int_{K \cap F} f \leq \left( c \sqrt{\frac n{k \log(n)} } \right)^{-k}. \] As a consequence we obtain a lower bound for $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ with $K$ a convex body, complementing the upper bound in \cite{koldobsky2011isomorphic}. This is \[c \sqrt{n/k} (\log(n))^{-1/2} \leq \sup_K d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}.\] The case $k=1$ was obtained previously in [5,6]. |
| title | The lower dimensional slicing inequality for functions and related distance inequalities |
| topic | Metric Geometry 52A20, 44A12 |
| url | https://arxiv.org/abs/2405.10223 |