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.
Read more about this topic: Filling Radius
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