A quantitative version of the transversality theorem

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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