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'':
:$\backslash mathrm(C\backslash subset\; \backslash mathbb^2)\; =\; R.$

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 $\backslash varepsilon$-neighborhoods of the loop ''C'', denoted :$U\_\backslash varepsilon\; C\; \backslash subset\; \backslash mathbb^2.$ As $\backslash varepsilon>0$ increases, the $\backslash varepsilon$-neighborhood $U\_\backslash varepsilon\; C$ 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 $\backslash mathrm(C\backslash subset\; \backslash mathbb^2)$ to be the infimum of $\backslash varepsilon\; >\; 0$ such that the loop ''C'' contracts to a point in $U\_\backslash varepsilon\; C$. 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 $U\_\backslash varepsilon\; X\backslash subset\; E$ 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 $\backslash mathbb$ or $\backslash mathbb\_2$, depending on whether or not ''X'' is orientable. Then the fundamental class, denoted ''', of a compact ''n''-dimensional manifold ''X'', is a generator of the homology group $H\_n(X;A)\backslash simeq\; A$, and we set :$\backslash mathrm(X\backslash subset\; E)\; =\; \backslash inf\; \backslash left\backslash ,$ where $\backslash iota\_\backslash varepsilon$ 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 $L^\backslash infty(X)$ of bounded Borel functions on ''X'', equipped with the sup norm $\backslash |\backslash cdot\backslash |$. Namely, we map a point $x\backslash in\; X$ to the function $f\_x\backslash in\; L^\backslash infty(X)$ defined by the formula $f\_x(y)\; =\; d(x,y)$ for all $y\backslash in\; X$, where ''d'' is the distance function defined by the metric. By the triangle inequality we have $d(x,y)\; =\; \backslash |\; f\_x\; -\; f\_y\; \backslash |,$ 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 $E=\; L^\backslash infty(X)$ in the formula above, and define :$\backslash mathrm(X)=\backslash mathrm\; \backslash left(\; X\backslash subset\; L^(X)\; \backslash right).$

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, *:$\backslash mathrm\; M\backslash ge\; \backslash frac.$

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. * {{Systolic geometry navbox Category:Riemannian geometry Category:Differential geometry Category:Systolic geometry

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 $\backslash varepsilon$-neighborhoods of the loop ''C'', denoted :$U\_\backslash varepsilon\; C\; \backslash subset\; \backslash mathbb^2.$ As $\backslash varepsilon>0$ increases, the $\backslash varepsilon$-neighborhood $U\_\backslash varepsilon\; C$ 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 $\backslash mathrm(C\backslash subset\; \backslash mathbb^2)$ to be the infimum of $\backslash varepsilon\; >\; 0$ such that the loop ''C'' contracts to a point in $U\_\backslash varepsilon\; C$. 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 $U\_\backslash varepsilon\; X\backslash subset\; E$ 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 $\backslash mathbb$ or $\backslash mathbb\_2$, depending on whether or not ''X'' is orientable. Then the fundamental class, denoted ''', of a compact ''n''-dimensional manifold ''X'', is a generator of the homology group $H\_n(X;A)\backslash simeq\; A$, and we set :$\backslash mathrm(X\backslash subset\; E)\; =\; \backslash inf\; \backslash left\backslash ,$ where $\backslash iota\_\backslash varepsilon$ 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 $L^\backslash infty(X)$ of bounded Borel functions on ''X'', equipped with the sup norm $\backslash |\backslash cdot\backslash |$. Namely, we map a point $x\backslash in\; X$ to the function $f\_x\backslash in\; L^\backslash infty(X)$ defined by the formula $f\_x(y)\; =\; d(x,y)$ for all $y\backslash in\; X$, where ''d'' is the distance function defined by the metric. By the triangle inequality we have $d(x,y)\; =\; \backslash |\; f\_x\; -\; f\_y\; \backslash |,$ 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 $E=\; L^\backslash infty(X)$ in the formula above, and define :$\backslash mathrm(X)=\backslash mathrm\; \backslash left(\; X\backslash subset\; L^(X)\; \backslash right).$

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, *:$\backslash mathrm\; M\backslash ge\; \backslash frac.$

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. * {{Systolic geometry navbox Category:Riemannian geometry Category:Differential geometry Category:Systolic geometry