Extremal problems for surfaces with bounded absolute (total) mean integral curvature in $n$-dimensionai space

Some inequalities are proved which relate the absolute mean integral curvature of hypersurface in $n$-dimensional Euclidean space with the volume and diameter of $n$-dimensional body are proved. Lemma of minimality of measure of $(n-1)$-dimenstonal planes set is the focus of attention: hypersphere as the element of set of closed hypersurfaces, bounding the body of fixed volume, has this property.