The mapping properties of filling radius and packing radius and their applications

Let f:V→W be a map between closed, oriented Riemannian n-manifolds. It is shown that FillRad(W)⩽dil(f)⋅FillRad(V), if |deg(f)|=1. By this mapping property, we obtain an estimate from below for the filling radius of a closed, oriented, nonpositively curved manifold, or a manifold with sectional curvature bounded above by a positive constant. In addition, a similar mapping property of packing radius and a corollary are also obtained.