$\mathbf{C^2}$-Lusin approximation of strongly convex bodies
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913539784966144 |
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| author | Azagra, Daniel Drake, Marjorie Hajłasz, Piotr |
| author_facet | Azagra, Daniel Drake, Marjorie Hajłasz, Piotr |
| contents | We prove that, if $W \subset \mathbb{R}^n$ is a locally strongly convex body (not necessarily compact), then for any open set $V \supset \partial W$ and $\varepsilon>0$, and $V \supset \partial W$ is open, then there exists a $C^2$ locally strongly convex body $W_{\varepsilon, V}$ such that $\mathcal{H}^{n-1}(\partial W_{\varepsilon, V}\triangle\,\partial W)<\varepsilon$ and $\partial W_{\varepsilon, V}\subset V$. Moreover, if $W$ is strongly convex, then $W_{\varepsilon, V}$ is strongly convex as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathbf{C^2}$-Lusin approximation of strongly convex bodies Azagra, Daniel Drake, Marjorie Hajłasz, Piotr Classical Analysis and ODEs Differential Geometry We prove that, if $W \subset \mathbb{R}^n$ is a locally strongly convex body (not necessarily compact), then for any open set $V \supset \partial W$ and $\varepsilon>0$, and $V \supset \partial W$ is open, then there exists a $C^2$ locally strongly convex body $W_{\varepsilon, V}$ such that $\mathcal{H}^{n-1}(\partial W_{\varepsilon, V}\triangle\,\partial W)<\varepsilon$ and $\partial W_{\varepsilon, V}\subset V$. Moreover, if $W$ is strongly convex, then $W_{\varepsilon, V}$ is strongly convex as well. |
| title | $\mathbf{C^2}$-Lusin approximation of strongly convex bodies |
| topic | Classical Analysis and ODEs Differential Geometry |
| url | https://arxiv.org/abs/2407.05745 |