Measuring the convexity of compact sumsets with the Schneider non-convexity index
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| Format: | Preprint |
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2024
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| _version_ | 1866911861146910720 |
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| author | Meyer, Mark |
| author_facet | Meyer, Mark |
| contents | In recent work, Franck Barthe and Mokshay Madiman introduced the concept of the Lyusternik region, denoted by $Λ_{n}(m)$, to better understand volumes of sumsets. They gave a characterization of $Λ_{n}(2)$ (the volumes of compact sets in $\mathbb{R}^n$ when at most $m=2$ sets are added together) and proved that Lebesgue measure satisfies a fractional superadditive property. We attempt to imitate the idea of the Lyusternik region by defining a region based on the Schneider non-convexity index function, which was originally defined by Rolf Schneider in 1975. We call this region the Schneider region, denoted by $S_{n}(m)$. In this paper, we will give an initial characterization of the region $S_{1}(2)$ and in doing so, we will prove that the Schneider non-convexity index of a sumset $c(A_1+A_2)$ has a best lower bound in terms of $c(A_1)$ and $c(A_2)$. We will pose some open questions about extending this lower bound to higher dimensions and large sums. We will also show that, analogous to Lebesgue measure, the Schneider non-convexity index has a fractional subadditive property. Regarding the Lyusternik region, we will show that when the number of sets being added is $m\geq3$, that the region $Λ_{n}(m)$ is not closed, proving a new qualitative property for the region. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_00221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measuring the convexity of compact sumsets with the Schneider non-convexity index Meyer, Mark Metric Geometry Combinatorics In recent work, Franck Barthe and Mokshay Madiman introduced the concept of the Lyusternik region, denoted by $Λ_{n}(m)$, to better understand volumes of sumsets. They gave a characterization of $Λ_{n}(2)$ (the volumes of compact sets in $\mathbb{R}^n$ when at most $m=2$ sets are added together) and proved that Lebesgue measure satisfies a fractional superadditive property. We attempt to imitate the idea of the Lyusternik region by defining a region based on the Schneider non-convexity index function, which was originally defined by Rolf Schneider in 1975. We call this region the Schneider region, denoted by $S_{n}(m)$. In this paper, we will give an initial characterization of the region $S_{1}(2)$ and in doing so, we will prove that the Schneider non-convexity index of a sumset $c(A_1+A_2)$ has a best lower bound in terms of $c(A_1)$ and $c(A_2)$. We will pose some open questions about extending this lower bound to higher dimensions and large sums. We will also show that, analogous to Lebesgue measure, the Schneider non-convexity index has a fractional subadditive property. Regarding the Lyusternik region, we will show that when the number of sets being added is $m\geq3$, that the region $Λ_{n}(m)$ is not closed, proving a new qualitative property for the region. |
| title | Measuring the convexity of compact sumsets with the Schneider non-convexity index |
| topic | Metric Geometry Combinatorics |
| url | https://arxiv.org/abs/2405.00221 |