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

The classical Cauchy surface area formula states that the surface area of the boundary $\partial K=\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 Σ along all directions in $\mathbb{S}^{n-1}$ . In this note, we generalize the formula to the boundary of arbitrary n-dimensional submanifold in $\mathbb{R}^{n}$ by introducing a natural notion of projected areas along any direction in $\mathbb{S}^{n-1}$ . This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013) by using a tubular neighborhood. We also define the projected r-volumes of Σ onto any r-dimensional subspaces and obtain a recursive formula for mean projected r-volumes of Σ.