The Dyn-Farkhi conjecture and the convex hull of a sumset in two dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910530356117504 |
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| author | Meyer, Mark |
| author_facet | Meyer, Mark |
| contents | For a compact set $A$ in $\mathbb{R}^n$ the Hausdorff distance from $A$ to $\text{conv}(A)$ is defined by
\begin{equation*}
d(A):=\sup_{a\in\text{conv}(A)}\inf_{x\in A}|x-a|,
\end{equation*}
where for $x=(x_1,\dots,x_n)\in\mathbb{R}^n$ we use the notation $|x|=\sqrt{x_1^2+\dots+x_n^2}$. It was conjectured in 2004 by Dyn and Farkhi that $d^2$ is subadditive on compact sets in $\mathbb{R}^n$. In 2018 this conjecture was proved false by Fradelizi et al. when $n\geq3$. The conjecture can also be verified when $n=1$. In this paper we prove the conjecture when $n=2$ and in doing so we prove an interesting representation of the sumset $\text{conv}(A)+\text{conv}(B)$ for full dimensional compact sets $A,B$ in $\mathbb{R}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dyn-Farkhi conjecture and the convex hull of a sumset in two dimensions Meyer, Mark Metric Geometry For a compact set $A$ in $\mathbb{R}^n$ the Hausdorff distance from $A$ to $\text{conv}(A)$ is defined by \begin{equation*} d(A):=\sup_{a\in\text{conv}(A)}\inf_{x\in A}|x-a|, \end{equation*} where for $x=(x_1,\dots,x_n)\in\mathbb{R}^n$ we use the notation $|x|=\sqrt{x_1^2+\dots+x_n^2}$. It was conjectured in 2004 by Dyn and Farkhi that $d^2$ is subadditive on compact sets in $\mathbb{R}^n$. In 2018 this conjecture was proved false by Fradelizi et al. when $n\geq3$. The conjecture can also be verified when $n=1$. In this paper we prove the conjecture when $n=2$ and in doing so we prove an interesting representation of the sumset $\text{conv}(A)+\text{conv}(B)$ for full dimensional compact sets $A,B$ in $\mathbb{R}^2$. |
| title | The Dyn-Farkhi conjecture and the convex hull of a sumset in two dimensions |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2407.07033 |