Filling radius¶
In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X.
Core Idea¶
Filling radius is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. 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.
Scope of Application¶
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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.
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Dual definition via neighborhoods. As \varepsilon>0 increases, the \varepsilon -neighborhood U\varepsilon C swallows up more and more of the interior of the loop.
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Homological definition. One imbeds X in the Banach space L^\infty(X) of bounded Borel functions on X, equipped with the sup norm |\cdot| .
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Homological definition. Namely, we map a point x\in X to the function fx\in L^\infty(X) defined by the formula fx(y) = d(x,y).
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Homological definition. for all y\in X , where d is the distance function defined by the metric.
Clarity¶
A clear use of Filling radius names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X.
Manages Complexity¶
Filling radius compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—as \varepsilon>0 increases, the \varepsilon -neighborhood U\varepsilon C swallows up more and more of the interior of the loop.—and the practical consequence—this follows by combining the diameter upper bound mentioned above with Gromov's lower bound in terms of the systole (Gromov, 1983).
Abstract Reasoning¶
- Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X.
- Check operation and conditions. Denote by A the coefficient ring \mathbb{Z} or \mathbb{Z}2 , depending on whether or not X is orientable.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Filling radius transfers literally when a new case preserves the same carrier type, relation, and recognition test. 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. As \varepsilon>0 increases, the \varepsilon -neighborhood U\varepsilon C swallows up more and more of the interior of the loop. Beyond the home domain. No canonical parent is asserted for Filling radius.
Neighborhood in Abstraction Space¶
Filling radius sits in a crowded region of the domain-specific corpus (15th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Named Analytic Theorems & Operators (39 abstractions)
Nearest neighbors
- Julia set — 0.92
- S-procedure — 0.91
- Big O in probability notation — 0.91
- Rooted product of graphs — 0.91
- Single Vegetative Obstruction Model — 0.90
Computed from structural-signature embeddings · 2026-10-08