  We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which
relates the geometry of a submanifold to the measure of the geodesics in the
ambient space which it intersects. For hypersurfaces, this gives an extension
of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive
curvature. We also prove a modified version of Croke's isoperimetric inequality
for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp,
quantitative version of an isoperimetric inequality of Yau in spaces of
negative curvature. We discuss the relationship between these results, and we
develop several facts about the spaces of geodesics in Cartan-Hadamards
manifold that may be of independent interest.
