Relation To Diameter and Systole
The exact value of the filling radius is known in very few cases. A general inequality relating the filling radius and the Riemannian diameter of X was proved in (Katz, 1983): the filling radius is at most a third of the diameter. In some cases, this yields the precise value of the filling radius. Thus, the filling radius of the Riemannian circle of length 2π, i.e. the unit circle with the induced Riemannian distance function, equals π/3, i.e. a sixth of its length. This follows by combing the diameter upper bound mentioned above with Gromov's lower bound in terms of the systole (Gromov, 1983). More generally, the filling radius of real projective space with a metric of constant curvature is a third of its Riemannian diameter, see (Katz, 1983). Equivalently, the filling radius is a sixth of the systole in these cases. The precise value is also known for the n-spheres (Katz, 1983).
The filling radius is linearly related to the systole of an essential manifold M. Namely, the systole of such an M is at most six times its filling radius, see (Gromov, 1983). The inequality is optimal in the sense that the boundary case of equality is attained by the real projective spaces as above.
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