arXiv
Open Access
2026
Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics
Hanwen Liu
Abstrak
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.
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Hanwen Liu
Akses Cepat
Informasi Jurnal
- Tahun Terbit
- 2026
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