A surface area formula for compact hypersurfaces in R^n

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