A surface area formula for compact hypersurfaces in Rⁿ

The classical Cauchy surface area formula states that the surface area of the boundary ∂ K=Σ ∂ K = Σ of any n -dimensional convex body in the n -dimensional Euclidean space Rⁿ R n can be obtained by the average of the projected areas of Σ along all directions in Sⁿ⁻¹ S n − 1 . In this note, we generalize the formula to the boundary of arbitrary n -dimensional submanifold in Rⁿ R n by introducing a natural notion of projected areas along any direction in Sⁿ⁻¹ 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.