Geometric integral formulas of cylinders within slabs

Abstract

We present new expressions for the integrals of mean curvature of domains in $R^n$ by means of sections with cylinders. Then, we combine these expressions with the corresponding version of the invariant density of affine subspaces in $R^n$, in order to obtain pseudo-rotational formulae for all the integrals of mean curvature of ∂K. As particular cases, we present pseudo-rotational integral formulas for the volume, area, integral of mean curvature, and Euler-Poincaré characteristic of a connected domain of $R^3$, whose boundary is a surface, considering slabs in $R^3$ whose central plane passes through a fixed point, and cylinders contained in these slabs.