Filling radius
In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form. The filling radius of a simple loop C in the plane is defined as the largest radius, R > 0, of a circle that fits inside C:

Dual definition via neighborhoods

There is a kind of a dual point of view that allows one to generalize this notion in an extremely fruitful way, as shown by Gromov. Namely, we consider the -neighborhoods of the loop C, denoted

As  increases, the -neighborhood  swallows up more and more of the interior of the loop. The last point to be swallowed up is precisely the center of a largest inscribed circle. Therefore, we can  reformulate the above definition by defining 
 to  be the infimum of  such that the loop C contracts to a point in . Given a compact manifold X imbedded in, say, Euclidean space E, we could define the filling radius relative to the imbedding, by minimizing the size of the neighborhood  in which X could be homotoped to something smaller dimensional, e.g., to a lower-dimensional polyhedron. Technically it is more convenient to work with a homological definition. Homological definition

Denote by A the coefficient ring  or , depending on whether or not X is orientable. Then the fundamental class,  denoted [X], of a compact n-dimensional manifold X, is a generator of the homology group , and we set 

where  is the inclusion homomorphism. To define an absolute filling radius in a situation where X is equipped with a Riemannian metric g, Gromov proceeds as follows. One exploits Kuratowski embedding. One imbeds X in the Banach space  of bounded Borel functions on X, equipped with the sup norm . Namely, we map a point  to the function  defined by the formula 
for all , where d is the distance function defined by the metric. By the triangle inequality we have  and therefore the imbedding is strongly isometric, in the precise sense that internal distance and ambient distance coincide. Such a strongly isometric imbedding is impossible if the ambient space is a Hilbert space, even when X is the Riemannian circle (the distance between opposite points must be
, not 2!). We then set  in the formula above, and define

Properties

 The filling radius is at most a third of the diameter (Katz, 1983). 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 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 combining the diameter upper bound mentioned above with Gromov's lower bound in terms of the systole (Gromov, 1983)
The systole of  an essential manifold 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. The injectivity radius of compact manifold gives a lower bound on filling radius. Namely,

See also

Filling area conjecture
Gromov's systolic inequality for essential manifolds

References

 Gromov, M.: Filling Riemannian manifolds, Journal of Differential Geometry 18 (1983), 1–147. Katz, M.: The filling radius of two-point homogeneous spaces. Journal of Differential Geometry 18, Number 3 (1983), 505–511. Riemannian geometry
Differential geometry
Systolic geometry