  The paper investigates higher dimensional analogues of Burago's inequality
bounding the area of a closed surface by its total curvature. We obtain
sufficient conditions for hypersurfaces in 4-space that involve the Ricci
curvature. We get semi-local variants of the inequality holding in any
dimension that involve domains with non-vanishing Gauss-Kronecker curvature.
The paper also contains inequalities of isoperimetric type involving the total
curvature, as well as a "reverse" isoperimetric inequality for spaces with
constant curvature.
