---
title: Second Cheeger Constant
url: https://www.emergentmind.com/topics/second-cheeger-constant
type: topic
---

# Second Cheeger Constant

“Second Cheeger constant” is used for several inequivalent objects. In the higher Cheeger problem for domains, it usually denotes the two-cluster quantity
\[
h_2(\Omega)=\inf\Big\{\max\{h_1(E_1),h_1(E_2)\}: E_1,E_2\subset \Omega,\ |E_1|,|E_2|>0,\ E_1\cap E_2=\emptyset\Big\},
\]
while in the anisotropic setting the analogous invariant is
\[
h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.
\]
A different recent graph-theoretic usage arises from the study of the second largest spectral gap from \(1\) of the normalized Laplacian: the natural geometric quantity there is a new two-step constant \(\tilde h\), and the paper introducing it explicitly notes that “second Cheeger constant” is not its standard name [1706.07282] [2503.23092] [2606.08061].

## 1. Terminological scope

The phrase has to be interpreted from context. In continuum higher-order Cheeger theory, Bobkov and Parini define a full sequence \(h_k(\Omega)\), and the case \(k=2\) is literally the second Cheeger constant [1706.07282]. In the anisotropic/Finsler setting, the corresponding object is the second anisotropic Cheeger constant \(h_{2,F}\) [2503.23092]. By contrast, the 2026 graph paper on the second largest spectral gap from \(1\) introduces a new Cheeger-type constant \(\tilde h\) tailored to a two-step operator and presents it as the natural object if one wants a geometric constant associated to that spectral quantity, while also emphasizing that it is not the standard higher-order multiway Cheeger constant [2606.08061].

| Setting | Quantity | Defining role |
|---|---|---|
| Euclidean higher Cheeger theory | \(h_2(\Omega)\) | minimization of the worse \(h_1\)-value over two disjoint subsets |
| Anisotropic higher Cheeger theory | \(h_{2,F}(\Omega)\) | anisotropic perimeter/volume min-max over two disjoint subsets |
| Two-step graph theory for gaps from \(1\) | \(\tilde h\) | geometric constant associated to \(\tau^2\), the second largest eigenvalue of \((D^{-1}A)^2\) |

Several nearby notions are not “second Cheeger constants” in this sense. The dual Cheeger constant \(\bar h\) is a distinct invariant designed to control the top of the spectrum and near-bipartiteness rather than a higher-order partition problem [1207.3410]. Higher-dimensional discrete Cheeger theory for simplicial complexes studies a higher-dimensional analogue of the graph’s second eigenvalue side, not a second Cheeger constant [1401.2290]. The average-case \(k\)-fold constant of Kenter and Racliffe satisfies \(h_G^{(2)}=h_G\), so it does not single out a separate invariant called “the second Cheeger constant” [1501.01741].

## 2. The two-step graph constant associated with the second largest spectral gap from \(1\)

For a finite, simple graph \(G=(V,E)\) with no isolated vertices, the normalized Laplacian in the 2026 formulation is
\[
L:=L(G):=\id-D^{-1}A,
\]
with
\[
D:=\diag(\deg v_1,\ldots,\deg v_n),
\qquad
0=\lambda_1\le \lambda_2\le \cdots\le \lambda_n\le 2.
\]
Because \(|1-\lambda_1|=1\) is always the trivial largest distance from \(1\), the quantity of interest is
\[
\tau:=\max_{i\neq 1}|1-\lambda_i|,
\]
which the paper calls the “second largest spectral gap from \(1\).” A basic structural identity is
\[
\tau=\max\{1-\lambda_2,\lambda_n-1\}.
\]
Thus \(\tau\) simultaneously reflects poor expansion at the bottom of the spectrum and near-bipartiteness at the top [2606.08061].

The spectral backbone is the two-step operator
\[
M:=(\id-L)^2=(D^{-1}A)^2,
\]
whose eigenvalues are \((1-\lambda_i)^2\). With the weighted inner product
\[
\langle f,g\rangle:=\sum_{v\in V}\deg v\cdot f(v)\cdot g(v),
\]
the Rayleigh quotient is
\[
\RQ(f):=\frac{\langle Mf,f\rangle}{\langle f,f \rangle}
=\frac{\sum\limits_{w\in V}\frac{1}{\deg w}\left(\sum\limits_{v\in \mathcal{N}(w)}f(v)\right)^2}{\sum\limits_{w\in V} \deg w\cdot f(w)^2}.
\]
Since the largest eigenvalue of \(M\) is \(1=(1-\lambda_1)^2\), corresponding to constants, \(\tau^2\) is the second largest eigenvalue of \(M\), and
\[
\tau^2= \max_{\substack{f\in C(V)\setminus\{\mathbf{0}\}:\\\sum_{w\in V}\deg w\cdot f(w)=0} } \RQ(f).
\]

The corresponding geometric quantity is
\[
\tilde{h}(S):=\frac{1}{\vol S}\cdot \sum_{w\in V}\frac{e(w, S)^2}{\deg w},
\qquad
\tilde{h}:=\max_{\substack{\emptyset\neq S \subseteq V:\\ \vol S\leq \vol V/2}} \tilde{h}(S),
\]
where
\[
\vol(S):=\sum_{v\in S}\deg v,
\qquad
e(w,S):=e(\{w\},S).
\]
For indicator functions,
\[
\RQ(\mathbbm{1}_S)=\tilde h(S).
\]
This identity is the direct analogue of the usual role of set indicators in classical Cheeger theory, but for the two-step operator \(M\). The same paper notes that this \(\tilde h\) is the primary candidate for what one might informally call a “second Cheeger constant” in this particular spectral problem, and explicitly distinguishes it from higher-order multiway Cheeger constants [2606.08061].

## 3. Relation to classical and dual Cheeger constants

The classical Cheeger constant and the dual Cheeger constant are
\[
h:=h(G):=\min_{\emptyset\neq S\subsetneq V}\frac{e(S,\overline{S})}{\min\{\vol(S),\vol(\overline{S})\}},
\]
and
\[
\bar{h}:= \bar{h}(G):=\max_{\substack{\text{partitions}\ V=S_1\sqcup S_2 \sqcup S_3\, :\ S_1,S_2\neq\emptyset}}
\frac{2\cdot e(S_1,S_2)}{\vol(S_1)+\vol(S_2)}.
\]
They govern the bottom and top spectral edges through
\[
1-\sqrt{1-h^2} \leq \lambda_2 \leq 2h,
\qquad
2\overline h \leq \lambda_n \leq 1 + \sqrt{1-\left(1-\overline h\right)^2}.
\]
Since
\[
\tau=\max\{1-\lambda_2,\lambda_n-1\},
\]
the 2026 paper first derives a combined bound from
\[
\widehat{h}:=\min\left\{h,1-\bar h\right\},
\]
namely
\[
1-2\widehat{h}\leq \tau \leq \sqrt{1-\widehat{h}^2}.
\]
This shows that \(\tau\) cannot be described naturally by \(h\) alone or by \(\bar h\) alone [2606.08061].

The interpretation of \(\tilde h\) is genuinely two-step. For
\[
h(S)=\frac{e(S,\overline S)}{\vol(S)},
\]
\(h(S)\) is the probability that a one-step random walk started from \(S\) leaves \(S\), so \(1-h(S)\) is the probability that it stays in \(S\) after one step. By contrast,
\[
\tilde h(S)=\frac{1}{\vol S}\sum_{w\in V}\frac{e(w,S)^2}{\deg w}
\]
is the probability that a walk started in \(S\) returns to \(S\) after two steps, with the starting vertex in \(S\) sampled proportional to degree. The key point is that \(h\) detects one-step escape from a bottleneck, whereas \(\tilde h\) detects two-step return behavior [2606.08061].

The main sharp Cheeger-type inequalities are
\[
2\tilde{h}-1\leq \tau^2\leq \sqrt{1-(1-h)^2}.
\]
The lower bound
\[
\tau^2\ge 2\tilde h-1
\]
is the new inequality involving \(\tilde h\); the upper bound is sharp with equality if and only if \(G\) is bipartite or disconnected. The same paper also proves
\[
\tilde h\le 1,
\]
with equality if and only if \(G\) is bipartite or disconnected, and
\[
\tilde{h}\geq \max_{\substack{\emptyset\neq S \subseteq V:\\ \vol S\leq \vol V/2}} \frac{\vol S}{\vol \mathcal{N}(S)}.
\]
These facts place \(\tilde h\) between ordinary expansion and near-bipartite two-step return structure [2606.08061].

The dual Cheeger constant remains a distinct object. On infinite graphs, it is introduced to control the top of the spectrum, not to provide a higher-order or “second” Cheeger constant. In the loopless case, the asymptotic relation
\[
\bar h_\infty=0 \Longleftrightarrow h_\infty=1 \Longleftrightarrow \sigma^{\mathrm{ess}(\Gamma)}=\{1\}
\]
shows that \(\bar h\) is a complementary invariant rather than a two-cluster one [1207.3410].

## 4. The second Cheeger constant in the higher Cheeger problem

In the continuum higher Cheeger problem of Bobkov and Parini, the \(k\)-th Cheeger constant of a measurable set \(\Omega\subset\mathbb R^N\) is
\[
h_k(\Omega) := \inf \left\{ \max_{i=1,\dots,k} \frac{P(E_i)}{|E_i|}:~ E_i \subset \Omega,~ |E_i| > 0 ~ \forall i,~ E_i \cap E_j = \emptyset ~ \forall i \neq j \right\}.
\]
The specialization \(k=2\) gives the second Cheeger constant
\[
h_2(\Omega) = \inf\left\{ \max\left\{\frac{P(E_1)}{|E_1|},\frac{P(E_2)}{|E_2|}\right\} : E_1,E_2\subset\Omega,\ |E_1|,|E_2|>0,\ E_1\cap E_2=\emptyset \right\}.
\]
Equivalently,
\[
h_2(\Omega)=\inf\Big\{\max\{h_1(E_1),h_1(E_2)\}: E_1,E_2\subset \Omega \text{ measurable, pairwise disjoint}\Big\}.
\]
A minimizing pair is a Cheeger 2-tuple, or a pair of coupled Cheeger sets [1706.07282].

For bounded measurable \(\Omega\), existence holds in a strong adjusted form: there exists a \(2\)-adjusted Cheeger couple. The central equilibrium condition is the \(1\)-adjusted relation
\[
h_1(\Omega\setminus E_2)=h_1(E_1)=\frac{P(E_1)}{|E_1|},
\qquad
h_1(\Omega\setminus E_1)=h_1(E_2)=\frac{P(E_2)}{|E_2|}.
\]
This shows that each component is itself a Cheeger set of the complement of the other. The theory does not require
\[
E_1\cup E_2=\Omega,
\]
so a residual set \(\Omega\setminus(E_1\cup E_2)\) may remain.

For bounded open \(\Omega\), regularity of \(1\)-adjusted tuples is substantial. Each \(\partial^*E_i\cap\Omega\) is \(C^{1,\gamma}\) for every \(\gamma\in(0,\tfrac12)\), the singular set has Hausdorff dimension at most \(N-8\), and if \(N\le 7\) then \(\partial E_i\cap\Omega\) is \(C^{1,\gamma}\). The reduced free boundary has constant mean curvature
\[
\frac{h_1(E_i)}{N-1},
\]
while for a \(2\)-adjusted tuple the interface \(\partial^*(E_iE_j)\) has constant mean curvature. If \(c_{ij}\) denotes the interface curvature measured from inside \(E_i\) and \(c_{ij}\ge0\), then
\[
h_1(E_i)\ge h_1(E_j)
\quad\text{and}\quad
h_1(E_i)\ge c_{ij},
\]
and if \(h_1(E_i)>h_1(E_j)\), then
\[
h_1(E_i)=c_{ij}.
\]
These curvature relations are the geometric balance laws of the second Cheeger problem [1706.07282].

The spectral link is exact for the second \(p\)-Laplacian eigenvalue. If
\[
\mathfrak L_2(p;\Omega)=\lambda_2(p;\Omega),
\]
then
\[
\lim_{p\to1}\lambda_2(p;\Omega)=h_2(\Omega).
\]
More generally,
\[
\mathfrak L_k(p;\Omega)\ge \left(\frac{h_k(\Omega)}{p}\right)^p.
\]
For \(k=2\), the second Cheeger constant is therefore the \(p\to1\) limit of the second Dirichlet eigenvalue.

The planar theory includes explicit model domains. For a disk \(B\subset\mathbb R^2\),
\[
h_2(B)=h_1(\text{half-disk}),
\]
and the minimizing pair consists of the Cheeger sets of two half-disks. For the annulus
\[
\Omega=B_1\setminus \overline{B_R}\subset \mathbb R^2,
\]
one has
\[
h_2(\Omega)=h_1(\Omega')=\min_{r\in[0,\frac{1-R}{2}]}\mathcal F(r),
\]
where \(\Omega'\) is a half-ring; the minimizing pair is unique up to rotation and consists of two opposite copies of the unique half-ring Cheeger set [1706.07282].

## 5. The anisotropic second Cheeger constant

In the anisotropic setting, one fixes an even, convex, positively \(1\)-homogeneous \(C^2\) integrand \(F\), uniformly equivalent to the Euclidean norm, and defines the anisotropic perimeter
\[
P_F(E;\Omega):=|D\chi_E|_F(\Omega)
=\sup\left\{\int_E \operatorname{div}\sigma\,dx:\ \sigma\in C_c^\infty(\Omega;\mathbb R^n),\ F^\circ(\sigma)\le 1\right\}.
\]
For Lipschitz boundary,
\[
P_F(E;\Omega)=\int_{\partial E\cap \Omega} F(\nu_E)\,d\mathcal H^{n-1}.
\]
The first anisotropic Cheeger constant is
\[
h_{1,F}(\Omega):=\inf_{E\subset \Omega,\ |E|>0}\frac{P_F(E)}{|E|},
\]
and the second anisotropic Cheeger constant is
\[
h_{2,F}(\Omega):=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.
\]
Equivalently,
\[
h_{2,F}(\Omega) = \inf\left\{\max\{h_{1,F}(E_1),h_{1,F}(E_2)\}:\ E_1,E_2\subset\Omega,\ E_1\cap E_2=\emptyset,\ |E_i|>0\right\}.
\]
The infimum is attained by connected sets \(C_1,C_2\subset\Omega\), called a pair of coupled anisotropic Cheeger sets [2503.23092].

The associated spectral operator is the anisotropic \(p\)-Laplacian
\[
-Q_pu:=-\operatorname{div}\big(F^{p-1}(\nabla u)\,F_\xi(\nabla u)\big).
\]
Its second Dirichlet eigenvalue satisfies a domain decomposition formula:
\[
\lambda_{2,F}(p,\Omega) = \inf\Big\{ \max\{\lambda_{1,F}(p,\Omega_1),\lambda_{1,F}(p,\Omega_2)\}:\  \Omega_1,\Omega_2\subset\Omega,\ \Omega_1\cap\Omega_2=\emptyset \Big\}.
\]
This is the precise spectral counterpart of the min-max definition of \(h_{2,F}\). The paper proves the anisotropic second Cheeger inequality
\[
\lambda_{2,F}(p,\Omega)\ge \left(\frac{h_{2,F}(\Omega)}{p}\right)^p,
\]
and the exact asymptotic identification
\[
\lim_{p\to1^+}\lambda_{2,F}(p,\Omega)=h_{2,F}(\Omega)
\qquad\text{for bounded Lipschitz open }\Omega.
\]

A key geometric feature is the nodal-domain mechanism. If \(u_{2,p}\) is a second eigenfunction with nodal domains \(\Omega_p^+\) and \(\Omega_p^-\), then
\[
\lim_{p\to1^+}\max\{h_{1,F}(\Omega_p^+),h_{1,F}(\Omega_p^-)\} = h_{2,F}(\Omega).
\]
Thus the positive and negative nodal domains asymptotically realize the two competing Cheeger pieces. This is why the second eigenvalue is special: the paper notes that for \(k\ge3\) one has only
\[
\limsup_{p\to1^+}\lambda_{k,F}(p,\Omega)\le h_{k,F}(\Omega),
\]
whereas for \(k=2\) the nodal decomposition aligns exactly with the two-set Cheeger problem.

The anisotropic theory also includes comparison principles. One has
\[
h_{2,F}(\Omega)\ge h_{1,F}(\Omega),
\]
with strict inequality when \(\Omega\) has a unique anisotropic Cheeger set, and an anisotropic Hong–Krahn–Szegő bound
\[
h_{2,F}(\Omega)\ge h_{2,F}(W),
\]
where \(W\) is the union of two disjoint Wulff shapes, each of measure \(|\Omega|/2\). The paper further proves existence of a \(1\)-adjusted anisotropic Cheeger couple satisfying
\[
h_{1,F}(\Omega\setminus E_2)=h_{1,F}(E_1)=\frac{P_F(E_1)}{|E_1|},
\qquad
h_{1,F}(\Omega\setminus E_1)=h_{1,F}(E_2)=\frac{P_F(E_2)}{|E_2|}.
\]
This is the anisotropic analogue of the adjustment structure in the Euclidean higher Cheeger problem [2503.23092].

## 6. Distinct but adjacent notions

The dual Cheeger constant on graphs is not a second Cheeger constant. In the infinite-graph theory,
\[
\bar h(\Omega) = \sup_{\substack{V_1,V_2\subset \Omega\\ V_1,V_2\neq\emptyset,\ V_1\cap V_2=\emptyset}}
\frac{2|E(V_1,V_2)|}{\mathrm{vol}(V_1)+\mathrm{vol}(V_2)},
\]
and it is designed to capture closeness to bipartiteness and to control the top eigenvalue through inequalities such as
\[
2\bar h(\Omega)+h(\Omega)\le \lambda_{\max}(\Omega) \le 1+\sqrt{1-(1-\bar h(\Omega))^2}.
\]
The paper introducing \(\bar h\) explicitly states that it is not a second or higher-order Cheeger constant [1207.3410].

Higher-dimensional discrete Cheeger theory for simplicial complexes is also different. Parzanchevski–Rosenthal–Tessler’s combinatorial Cheeger constant \(h(X)\) and the later extension to arbitrary complexes concern the higher-dimensional analogue of the graph spectral gap:
\[
\lambda(X):=\min\operatorname{Spec}\!\left(L_{k-1}^{\mathrm{up}(X)}\big|_{(B^{k-1})^\perp}\right),
\]
with
\[
\lambda(X)\le h(X).
\]
This is a higher-dimensional analogue of the graph’s second smallest Laplacian eigenvalue side, not a second Cheeger constant in the multiway sense [1401.2290].

The \(k\)-fold constant of Kenter and Racliffe is an average-case multiway invariant,
\[
h_G^{(k)}=\inf_{\mathcal S}\frac{1}{k}\sum_{i\ne j} \frac{e(S_i,S_j)}{\min\{\operatorname{Vol}(S_i),\operatorname{Vol}(S_j)\}},
\]
and it satisfies
\[
h_G^{(2)}=h_G.
\]
Accordingly, its \(k=2\) specialization is exactly the ordinary Cheeger constant, not a separate “second” constant [1501.01741].

Finally, sheaf-theoretic coboundary expansion on graphs generalizes the ordinary Cheeger constant via
\[
\cb_0(X,w,\underline{\mathbf F_2}_X)=h(X,w),
\]
but that theory does not define or analyze a second Cheeger constant, a multiway partition constant, or any two-cluster analogue [2208.01776].

Taken together, these lines of work support a precise terminological conclusion. In continuum partition theory, “second Cheeger constant” refers to the two-set min-max invariant \(h_2\), and in the anisotropic/Finsler setting to \(h_{2,F}\). In the recent graph theory of the second largest spectral gap from \(1\), the natural analogue is instead the two-step return constant
\[
\tilde h=\max_{\substack{\emptyset\neq S \subseteq V:\\ \vol S\leq \vol V/2}}
\frac{1}{\vol S}\sum_{w\in V}\frac{e(w,S)^2}{\deg w},
\]
which is a new Cheeger-type quantity rather than the standard higher-order one [1706.07282] [2503.23092] [2606.08061].

Source: https://www.emergentmind.com/topics/second-cheeger-constant