Relation between area and volume for λ-convex sets in Hadamard manifolds

It is known that for a sequence {Omega (t)} of convex sets expanding over the whole hyperbolic space Hn+1 the limit of the quotient vol(Omega (t))/vol(partial derivative Omega (t)) is less or equal than 1/n, and exactly 1/n when the sets considered are convex with respect to horocycles. When convexity is with respect to equidistant lines, i.e., curves with constant geodesic curvature lambda less than one, the above limit has lambda /n as lower bound. Looking how the boundary bends, in this paper we give bounds of the above quotient for a compact lambda -convex domain in a complete simply-connected manifold of negative and bounded sectional curvature, a Hadamard manifold. Then we see that the limit of vol(Omega (t))/vol(partial derivative Omega (t)) for sequences of lambda -convex domains expanding over the whole space lies between the values lambda /nk(2)(2) and 1/nk(1).