Cheeger Isoperimetric Constant

• The Cheeger isoperimetric constant is defined as the infimum ratio of boundary measure to enclosed volume, serving as a bottleneck measure in geometric, measure-theoretic, and graph settings.
• It plays a crucial role in spectral geometry by providing lower bounds on Laplace eigenvalues through Cheeger and Buser type inequalities.
• The constant connects isoperimetry, curvature, and asymptotic geometry, offering insights into spectral gaps, rigidity results, and comparison theorems across different mathematical structures.

The Cheeger isoperimetric constant is the infimum of a boundary-to-volume ratio and records the smallest possible ratio of boundary area to enclosed volume, or more generally boundary measure to enclosed mass. In Riemannian, metric-measure, graph, and measure-theoretic settings, it serves as a quantitative expression of the global isoperimetric bottleneck. Its central role is spectral: lower bounds on the Cheeger constant force lower bounds on the first nonzero Laplace eigenvalue or on the bottom of the spectrum, while converse inequalities and rigidity statements relate geometry, curvature, and asymptotic structure to sharp isoperimetric behavior (Gayet et al., 3 Mar 2025, Cavalletti et al., 2016, Bauer et al., 2012).

1. Core definitions and equivalent formulations

The precise definition depends on the ambient category, but the common structure is an infimum of “boundary size divided by enclosed mass.” On a compact Riemannian manifold $(M^n,g)$, one standard form is

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$

with the infimum taken over smooth domains with smooth boundary. On a non-compact Riemannian manifold, the usual form is

$h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$

where $U$ ranges over relatively compact open submanifolds with smooth boundary and compact closure. In metric-measure spaces $(X,d,m)$, the same idea is expressed through perimeter: $$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$ For a probability measure $\mu$ on $\mathbb R^n$, one may write

$$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$

where $\mu_+$ is the lower Minkowski content. On graphs, the classical combinatorial version is

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$0

while for weighted graphs with possibly unbounded degree one uses a $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$1-weighted edge boundary attached to an intrinsic metric (Gayet et al., 3 Mar 2025, Cavalletti et al., 2016, Martínez-Pérez et al., 2016, Liehr, 16 Feb 2026, Bauer et al., 2012).

Setting	Admissible sets	Formula
Compact Riemannian manifold	Smooth $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$2	$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$3
Non-compact Riemannian manifold	Relatively compact $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$4 with smooth boundary	$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$5
Metric-measure space	Borel $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$6 with perimeter $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$7	$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$8
Graph	Finite vertex set $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$9 or finite $h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$0	$h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$1 or $h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$2

A second standard formulation uses the isoperimetric profile. If

$h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$3

then

$h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$4

Similarly, for a manifold one may define $h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$5 and then recover

$h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$6

This formulation is particularly useful in existence, rigidity, and comparison arguments (Cavalletti et al., 2016, Pallete et al., 2023).

In weighted graph theory, Bauer–Keller–Wojciechowski introduced an intrinsic-metric version tailored to unbounded graph Laplacians. If $h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$7 is intrinsic for the graph $h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$8 over $h(M)\;=\;\inf_U\frac{\Vol_{n-1}(\partial U)}{\Vol_n(U)},$9, meaning

$U$0

and if

$U$1

then

$U$2

This reduces to the usual combinatorial constant in the special case $U$3, $U$4, and $U$5 on every edge (Bauer et al., 2012).

2. Fundamental spectral inequalities

The classical spectral statement is Cheeger’s inequality. For a compact Riemannian manifold,

$U$6

where $U$7 is the first nonzero eigenvalue of the positive Laplace–Beltrami operator. The same lower bound persists in perimeter-based metric-measure formulations and in non-smooth $U$8 spaces, where the proof proceeds through coarea estimates, median splitting, and Cauchy–Schwarz (Gayet et al., 3 Mar 2025, Ponti et al., 2019).

Upper bounds in the opposite direction are Buser-type inequalities. On a compact $U$9-manifold with $(X,d,m)$0, Buser proved

$(X,d,m)$1

A simplified proof replaces Buser’s Dirichlet-region lemma by a volume comparison in concentric balls of radii $(X,d,m)$2 and $(X,d,m)$3, together with a distance-to-hypersurface Riccati estimate for $(X,d,m)$4 (Charalambous et al., 2022).

In $(X,d,m)$5 spaces, De Ponti–Mondino obtained dimension-free sharp Buser inequalities. If $(X,d,m)$6, then

$(X,d,m)$7

and this is sharp because equality is realized on Gaussian space. If $(X,d,m)$8, one gets an optimized bound strictly smaller than $(X,d,m)$9, while for $$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$0,

$$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$1

The proof is semigroup-based and uses a sharp Bakry–Gentil–Ledoux-type gradient estimate for the heat flow (Ponti et al., 2019).

For unbounded graph Laplacians, the intrinsic-metric Cheeger constant yields genuinely nontrivial spectral control. If $$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$2 is the self-adjoint Laplacian associated with $$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$3, then

$$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$4

Under the extra assumptions $$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$5 and $$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$6 (or $$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$7) on every edge,

$$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$8

For the essential spectrum, defining

$$h(X)\;=\;\inf\{P(E)/m(E): E\subset X\ \text{Borel},\ 0<m(E)\le 1/2\}.$$9

one obtains

$\mu$0

These estimates remain nontrivial even when vertex degrees are unbounded (Bauer et al., 2012).

3. Discrete, graph-theoretic, and asymptotic variants

The graph-theoretic theory separates sharply into bounded-degree and unbounded-degree regimes. In the bounded-degree normalized setting, the intrinsic-metric constant agrees with the classical combinatorial Cheeger constant, and the classical normalized-Laplacian Cheeger inequality is recovered. Beyond the hypothesis $\mu$1, the $\mu$2-weighting of edges becomes essential to maintain a nontrivial lower bound on $\mu$3; Bauer–Keller–Wojciechowski exhibit antitrees of polynomial growth for which previous Cheeger-type bounds vanish, whereas their intrinsic formulation gives $\mu$4 and hence $\mu$5 (Bauer et al., 2012).

For Gromov-hyperbolic graphs and manifolds with bounded local geometry and a pole, positivity of the Cheeger constant is characterized by the visual boundary. If $\mu$6 is a uniform, proper, hyperbolic graph with a pole, then

$\mu$7

The analogous statement holds for complete noncompact hyperbolic manifolds with bounded local geometry and a pole. In this setting, $\mu$8 yields $\mu$9, solvability of the Dirichlet problem at infinity, and identification of the Martin boundary with the Gromov boundary (Martínez-Pérez et al., 2016).

In supercritical bond percolation on $\mathbb R^n$0, $\mathbb R^n$1, a modified Cheeger constant

$\mathbb R^n$2

captures the correct large-box asymptotics on the infinite open cluster. Gold proves that there is a deterministic norm $\mathbb R^n$3 and a Wulff crystal $\mathbb R^n$4 such that optimal subgraphs converge, after rescaling, to translates of $\mathbb R^n$5, and

$\mathbb R^n$6

almost surely, where

$\mathbb R^n$7

This settles a conjecture of Benjamini for the modified Cheeger constant used there (Gold, 2016).

A different discrete generalization is the Houdré–Tetali family $\mathbb R^n$8. For $\mathbb R^n$9, Houdré and Tetali conjectured a strengthened hard direction of Cheeger’s inequality, but Lau–Tjowasi show that the conjecture is false in general: the logarithmic factor in the Morris–Peres bound is necessary. At the same time, for every $$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$0 they prove

$$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$1

and the argument extends to directed graphs via Chung’s directed Laplacian (Lau et al., 2024).

4. Minimizers, regularity, and explicit computation

The Cheeger constant is tightly linked to isoperimetric minimizers. In the compact Riemannian case, geometric measure theory solves the isoperimetric problem for each prescribed volume $$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$2, and Buser showed that a minimizer $$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$3 can be chosen so that

$$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$4

The boundary is smooth away from an $$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$5 subset, and meets $$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$6 orthogonally if $$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$7 has boundary. For non-compact finite-area surfaces, existence is subtler because one must prevent loss of mass to infinity: if a Cheeger sequence has boundaries contained in a common compact set, then there exists a subset $$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$8 with

$$h_\mu=\inf_{0<\mu(A)<1}\frac{\mu_+(A)}{\min\{\mu(A),1-\mu(A)\}},$$9

and $\mu_+$0 is an embedded multicurve of constant geodesic curvature (Benson, 2015).

On geometrically finite hyperbolic surfaces, Adams–Morgan classify isoperimetric minimizers of fixed area: they are metric disks, horocusp neighborhoods, annuli bounded by parallel curves to a simple closed geodesic, or multigeodesic regions bounded by constant-distance curves parallel to several disjoint geodesics. Benson then shows that on any geometrically finite surface there is a one-dimensional Cheeger minimizer of positive length which is exactly the boundary of one of these candidates, and gives an algorithm for computing $\mu_+$1 from topological data, length spectrum information, mutual distances between simple closed geodesics, and the Euler characteristics of the complementary regions (Benson, 2015).

For planar domains, the geometry of minimizers can be completely explicit. If $\mu_+$2 is a Jordan domain with $\mu_+$3 and has no necks of radius $\mu_+$4, then its maximal Cheeger set is

$\mu_+$5

and $\mu_+$6 is the unique positive solution of the inner Cheeger formula

$\mu_+$7

This extends the classical union-of-balls characterization from convex planar sets to a much larger class of simply connected domains (Leonardi et al., 2017).

The measure-theoretic Cheeger problem can also admit exact solutions. For the Gaussian mixture

$\mu_+$8

the Cheeger constant is

$\mu_+$9

where $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$00 is defined by $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$01 and

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$02

All minimizers are half-spaces whose boundary is orthogonal to $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$03; uniqueness up to complement occurs exactly when $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$04 is a singleton (Liehr, 16 Feb 2026).

5. Curvature, comparison geometry, rigidity, and flow

Comparison geometry often turns the Cheeger constant into an asymptotic volume-growth invariant. For a complete, non-compact, properly immersed submanifold

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$05

in a manifold with a pole and an upper bound on radial sectional curvatures, Gimeno–Palmer compare extrinsic balls $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$06 with model balls $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$07 in a warped product $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$08. Under appropriate “balanced from below” or “balanced from above” hypotheses, they obtain upper and lower bounds for $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$09 in terms of

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$10

In particular, if $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$11 is minimal in a Cartan–Hadamard manifold with $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$12, then

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$13

The Euclidean minimal case gives $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$14, while for minimal surfaces in hyperbolic space with finite total extrinsic curvature one obtains

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$15

These estimates feed directly into spectral and probabilistic consequences (Gimeno et al., 2011).

Rigidity results show that the Cheeger constant can distinguish canonical metrics inside deformation spaces. For convex co-compact hyperbolic $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$16-manifolds that are quasi-Fuchsian or acylindrical,

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$17

where $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$18 is the unique solution of $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$19, and equality holds if and only if $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$20 is Fuchsian. The proof compares isoperimetric profiles with the totally geodesic model and uses that the renormalized volume is minimized exactly at the Fuchsian locus (Pallete et al., 2023).

Under Ricci flow on topological $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$21-spheres, the Cheeger constant is monotone but not strictly so. Williams proves that if $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$22 evolves by Ricci flow on $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$23, then $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$24 is a non-decreasing function of $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$25. He also constructs a smooth metric on $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$26 for which $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$27 at some time, showing that strict monotonicity fails in general (Williams, 2024).

In strictly negatively curved manifolds of infinite volume, the Cheeger constant also has a bounded-cohomological interpretation. If $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$28 has bounded geometry, then the following are equivalent: $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$29; the volume form admits a bounded primitive; and the bounded fundamental class $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$30 vanishes in bounded cohomology. Even without bounded geometry, positivity of $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$31 still implies vanishing of the bounded volume class (Hadziosmanovic, 27 Jul 2025).

6. Modern extensions and specialized directions

Recent work broadens the notion of Cheeger constant in several orthogonal directions. In metric-measure spaces satisfying the local curvature-dimension condition and essential non-branching, Cavalletti–Mondino prove the sharp perimeter isoperimetric inequality

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$32

and deduce

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$33

The one-dimensional localization argument also shows that, in this lower-Ricci regime, outer Minkowski content inequalities can be upgraded to the sharper perimeter formulation. Equality forces a spherical suspension structure (Cavalletti et al., 2016).

In random algebraic geometry, the Cheeger constant becomes a typical rather than worst-case invariant. For a random smooth plane curve $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$34 of degree $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$35 with the induced Fubini–Study metric,

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$36

and therefore

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$37

More generally, for random complex curves $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$38 inside projective manifolds, the paper proves a polynomial lower bound on $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$39 with overwhelming probability. Deterministically, by contrast, one can arrange arbitrarily short vanishing cycles and hence $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$40 (Gayet et al., 3 Mar 2025).

In Finsler geometry, a distinct “second Cheeger constant” is introduced: $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$41 For a forward complete non-compact Finsler metric-measure manifold with $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$42, the sharp isoperimetric inequality

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$43

holds for every bounded Borel set $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$44, where $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$45 is the volume entropy. Hence

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$46

Under uniform convexity and smoothness, this feeds into Cheeger–Buser type estimates for the bottom of the spectrum of the Finsler Laplacian (Cheng et al., 11 Jul 2025).

Higher-order variants arise for Steklov spectra. In finite spaces, measurable spaces, and compact Riemannian manifolds with boundary, the $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$47-th Cheeger–Steklov constant $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$48 is defined from a product of an interior boundary ratio and an exterior boundary ratio over $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$49 disjoint test regions. There exist universal constants $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$50 such that

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$51

with an improved logarithmic estimate for $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$52. For $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$53, this recovers the Cheeger-type lower bound for the first nonzero Steklov eigenvalue up to absolute constants (Hassannezhad et al., 2017).

Extremal problems for $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$54 itself have also become more precise. Under a minimal-width constraint for planar convex bodies,

$h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$55

and equality holds if and only if $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$56 is an equilateral triangle. The proof combines Ftouhi’s reverse Cheeger bound with sharp width–inradius and width–area inequalities, and a stability theorem controls Hausdorff-width asymmetry by the width–Cheeger deficit (Ftouhi et al., 26 Apr 2025).

Taken together, these results show that the Cheeger isoperimetric constant is not a single invariant with a single preferred definition, but a family of closely related bottleneck functionals whose exact form depends on whether the relevant boundary object is a smooth hypersurface, a finite-perimeter set, a weighted edge boundary, an outer Minkowski content, or a boundary-to-volume ratio adapted to a spectral problem. What remains stable across these settings is the structural role of $h(M)\;=\;\inf_{\Omega\subset M}\; \frac{\Vol_{n-1}(\partial\Omega)} {\min\{\Vol_n(\Omega),\Vol_n(M\setminus\Omega)\}},$57: it mediates between isoperimetry, spectral gap, curvature, and asymptotic geometry.