On the volume of convolution bodies in the plane
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913555354222592 |
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| author | Haddad, J. |
| author_facet | Haddad, J. |
| contents | For every convex body $K \subset \mathbb R^n$ and $δ\in (0,1)$, the $δ$-convolution body of $K$ is the set of $x \in \mathbb R^n$ for which $\left|K \cap (K+x)\right|_n \geq δ\left|K\right|_n$.
We show that for $n=2$ and any $δ\in (0,1)$, ellipsoids do not maximize the volume of the $δ$-convolution body of $K$, when $K$ runs over all convex bodies of a fixed volume.
This behavior is somehow unexpected and contradicts the limit case $δ\to 1^-$, which is governed by the Petty projection inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00212 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the volume of convolution bodies in the plane Haddad, J. Metric Geometry 52A10, 52A30 For every convex body $K \subset \mathbb R^n$ and $δ\in (0,1)$, the $δ$-convolution body of $K$ is the set of $x \in \mathbb R^n$ for which $\left|K \cap (K+x)\right|_n \geq δ\left|K\right|_n$. We show that for $n=2$ and any $δ\in (0,1)$, ellipsoids do not maximize the volume of the $δ$-convolution body of $K$, when $K$ runs over all convex bodies of a fixed volume. This behavior is somehow unexpected and contradicts the limit case $δ\to 1^-$, which is governed by the Petty projection inequality. |
| title | On the volume of convolution bodies in the plane |
| topic | Metric Geometry 52A10, 52A30 |
| url | https://arxiv.org/abs/2405.00212 |