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arXiv · 2603.21731

Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics

Hanwen Liu
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Abstract

We prove that for any nonlinear $f \in C^{1,\alpha}([0,1])$, the union of lines covering its graph has a Hausdorff dimension of at least $1+\alpha$, and this dimension bound is sharp. We then apply these geometric results to mathematical physics, proving that spacetime observability sets for conservation laws with $\alpha$-H\"older initial wave speeds possess a dimension of at least $\alpha$. 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. 2026-03-23. Hausdorff Dimension of Union of Lines Covering a Curve: Applications to Mathematical Physics. https://arxiv.org/abs/2603.21731

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