This article is about the Cheeger isoperimetric constant and Cheeger's inequality in Riemannian geometry. For a different use, see Cheeger constant (graph theory).
In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal area of a hypersurface that divides M into two disjoint pieces. In 1971, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on M to h(M). In 1982, Peter Buser proved a reverse version of this inequality, and the two inequalities put together are sometimes called the Cheeger-Buser inequality. These inequalities were highly influential not only in Riemannian geometry and global analysis, but also in the theory of Markov chains and in graph theory, where they have inspired the analogous Cheeger constant of a graph and the notion of conductance.
In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal...
a graph through the second eigenvalue of its Laplacian. The Cheegerconstant (also Cheeger number or isoperimetric number) of a graph is a numerical measure...
In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary...
typified by the Cheeger inequality which gives a relation between the first positive eigenvalue and an isoperimetric constant (the Cheegerconstant). Many versions...
this difference to other properties of the system. Cheegerconstant (graph theory) Cheegerconstant (Riemannian geometry) Eigengap Spectral gap (physics)...
expansion parameters. The edge expansion (also isoperimetric number or Cheegerconstant) h(G) of a graph G on n vertices is defined as h ( G ) = min 0 < |...
element of a chain complex, namely a set of vertices or a set of edges. Cheegerconstant See expansion. cherry A cherry is a path on three vertices. χ χ(G)...
property (τ) of Lubotzky–Zimmer. This can be taken to mean that the Cheegerconstant of the family of their Schreier coset graphs (with respect to a fixed...
about the structure of positively curved manifolds. The soul theorem (Cheeger & Gromoll 1972; Gromoll & Meyer 1969) implies that a complete non-compact...
related quantities include the Cheegerconstant of a Riemannian manifold and the (differently defined) Cheegerconstant of a graph. Berger, Marcel (2010)...
between 1/4 and 1 then M is diffeomorphic to a sphere. Cheeger's finiteness theorem. Given constants C, D and V, there are only finitely many (up to diffeomorphism)...
of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result...
isoperimetric dimension infinity. In fact the hyperbolic plane has positive Cheegerconstant. This means that it satisfies the inequality area ( ∂ D ) ≥ C vol...
There is a theorem proved by Jeff Cheeger, Kenji Fukaya and Mikhail Gromov, which states that: There exists a constant ε ( n ) {\displaystyle \varepsilon...
and is the key point in the proof of Gromov's compactness theorem.) The Cheeger–Gromoll splitting theorem states that if a complete Riemannian manifold...
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