The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $f\in \mathcal{ C}([0,1]^d,\mathbb{ R}^m)$ and a manifold $W\subset \mathbb{ R}^m$ of dimension $p$, a sharpness result on the upper quantitative estimate of the $(d+p-m)$-dimensional Hausdorff measure of the set $\mathcal{ Z}_{ W}^{ f}=\left\{ x\in [0,1]^d: f(x)\in W\right\}$, which was achieved in [8], will be proved in terms of power functions.
Categories: math.FA