A surface area formula for compact hypersurfaces in ${\mathbb{R}}^n$

The classical result of Cauchy’s surface area formula states that the surface area of the boundary $\mathrm{\partial }K=\mathrm{\Sigma }$ of any $n$-dimensional convex body in the $n$-dimensional Euclidean space ${\mathbb{R}}^n$ can be obtained by the average of the projected areas of $\mathrm{\Sigma }$ along all directions in ${\mathbb{S}}^{n-1}$. In this notes, we generalize the formula to the boundary of arbitrary $n$-dimensional submanifolds in ${\mathbb{R}}^n$ by defining a natural notion of projected areas along any direction in ${\mathbb{S}}^{n-1}$. This surface area formula derived from the new concept coincides with not only the result of the Crofton’s formula but that of De Jong \cite{de2013volume} by using tubular neighborhood. We also define the projected $r$-volumes of $\mathrm{\Sigma }$ onto any $r$-dimensional subspaces, and obtain a recursive formula for mean projected $r$-volumes of $\mathrm{\Sigma }$.