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: 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.
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
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.
Denote by A the coefficient ring
Then the fundamental class, denoted [X], of a compact n-dimensional manifold X, is a generator of the homology group
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
defined by the formula
, where d is the distance function defined by the metric.
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 π, not 2!).