Given an open limited set $E$ of $\mathbb{R}^n$ with smooth boundary, how could I subdivide this set $E$ in a finite number of normal sets? I remind you that the normal set is a set of this form, given an open set $A$ in $\mathbb{R}^{n-1}$ and two functions $\varphi_{1}$ and $\varphi_2$ defined on $A$ the corresponding normal set is $$\{(x,y) \in A : \varphi_{1}(x)<y<\varphi_{2}(x)\}.$$ Subdivide means to partition the set at most excluding $\mathcal{H}^{n}$ negligible sets. My question emerges from a proof of the Steiner inequality that Francesco Maggi gives in this book at pag 161. https://www.cambridge.org/core/books/sets-of-finite-perimeter-and-geometric-variational-problems/F8D0DABFFFB0D444C5AD5D37B3E3DBC1
What I know for certain is that I can choose a hyperplane $\Pi$ for which there are a $\mathcal{H}^{n-1}-$negligible amount of points with normal vector parallel to the hyperplane $\Pi.$ So almost all of the perpendicular lines to $\Pi$ are intersecting $E$ in a finite number of segments. In the figure I tried to give a sense of what I am asking.