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
Famous quotes containing the words dual and/or definition:
“Thee for my recitative,
Thee in the driving storm even as now, the snow, the winter-day
declining,
Thee in thy panoply, thy measur’d dual throbbing and thy beat
convulsive,
Thy black cylindric body, golden brass and silvery steel,”
—Walt Whitman (1819–1892)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)