A quantitative version of the transversality theorem

Andrew Murdza; Khai T. Nguyen

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arXiv · 2112.07107

A quantitative version of the transversality theorem

Andrew Murdza·Khai T. Nguyen

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Abstract

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $g\in \mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a global smooth manifold $W\subset \mathbb{R}^m$ of dimension $p$, we establish a quantitative estimate on the $(d+p-m)$-dimensional Hausdorff measure of the set $\mathcal{Z}_{W}^{g}=\left\{x\in [0,1]^d: g(x)\in W\right\}$. The obtained result is applied to quantify the total number of shock curves in weak entropy solutions to scalar conservation laws with uniformly convex fluxes in one space dimension.

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Andrew Murdza, Khai T. Nguyen. 2021-12-14. A quantitative version of the transversality theorem. https://arxiv.org/abs/2112.07107

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