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:

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 ${\displaystyle \varepsilon }$-neighborhoods of the loop C, denoted

As ${\displaystyle \varepsilon >0}$ increases, the ${\displaystyle \varepsilon }$-neighborhood ${\displaystyle U_{\varepsilon }C}$ 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 ${\displaystyle \mathrm {FillRad} (C\subset \mathbb {R} ^{2})}$ to be the infimum of ${\displaystyle \varepsilon >0}$ such that the loop C contracts to a point in ${\displaystyle U_{\varepsilon }C}$.

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