A Note on Generic Transversality of Euclidean Submanifolds

In this short note, we establish a quantitative description of the genericity of transversality of $C^1$-submanifolds in ${\mathbb{R}}^n$: Let $\Sigma \subset $\mathbb{R}$^n$bea$d$-dimensional$C^1$-embeddedsubmanifoldwhere$n \geq d+1$. Denote by \begin{equation} \mathscr{A}(\Sigma) := \bigg{ a \in $\mathbb{R}$^n: {\rm volume},\Big{ p\in\Sigma : \partial$\mathbb{B}$(a, |a-p|) $\text{ is not transversal to }\mathrm{\Sigma }\text{ at }p$ \Big} > 0 \bigg}. \end{equation} Then $\mathcal{A}(\mathrm{\Sigma })$ is contained in a countable union of $(n-d-1)$-dimensional affine planes.