Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914435191275520 |
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| author | Liu, Hanwen |
| author_facet | Liu, Hanwen |
| contents | We prove that for any nonlinear $f \in C^{1,α}([0,1])$, the union of lines covering its graph has a Hausdorff dimension of at least $1+α$, and this dimension bound is sharp. We then apply these geometric results to mathematical physics, proving that spacetime observability sets for conservation laws with $α$-Hölder initial wave speeds possess a dimension of at least $α$. Finally, we prove that if an absolutely integrable vector field $v$ on the boundary of a polyhedron exhibits a strictly positive total flux, then the union of the line field spanned by $v$ possesses a Hausdorff dimension of 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21731 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics Liu, Hanwen Analysis of PDEs Classical Analysis and ODEs We prove that for any nonlinear $f \in C^{1,α}([0,1])$, the union of lines covering its graph has a Hausdorff dimension of at least $1+α$, and this dimension bound is sharp. We then apply these geometric results to mathematical physics, proving that spacetime observability sets for conservation laws with $α$-Hölder initial wave speeds possess a dimension of at least $α$. Finally, we prove that if an absolutely integrable vector field $v$ on the boundary of a polyhedron exhibits a strictly positive total flux, then the union of the line field spanned by $v$ possesses a Hausdorff dimension of 3. |
| title | Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2603.21731 |