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

AbstractIt is known that for a sequence {Ωt} of convex sets expanding over the whole hyperbolic space Hn+1 the limit of the quotient vol(Ωt)/vol(∂Ωt) 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 λ less than one, the above limit has λ/n as lower bound. Looking how the boundary bends, in this paper we give bounds of the above quotient for a compact λ-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(Ωt)/vol(∂Ωt) for sequences of λ-convex domains expanding over the whole space lies between the values λ/nk22 and 1/nk1