Second Cheeger Constant

• Second Cheeger constant is a two-cluster min–max invariant defined by partitioning a domain into disjoint subsets, each evaluated via its first Cheeger ratio.
• In anisotropic settings, it is formulated using the anisotropic perimeter-to-volume ratio, leading to coupled Cheeger sets with sharp geometric balance laws.
• In graph theory, a related two-step constant quantifies the return probability in random walks, linking the second largest spectral gap from 1 to near-bipartiteness.

“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 (Bobkov et al., 2017, Piscitelli, 29 Mar 2025, Beers et al., 6 Jun 2026).

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 (Bobkov et al., 2017). In the anisotropic/Finsler setting, the corresponding object is the second anisotropic Cheeger constant $h_{2,F}$ (Piscitelli, 29 Mar 2025). 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 (Beers et al., 6 Jun 2026).

Setting	Quantity	Defining role
Euclidean higher Cheeger theory	$h_2(\Omega)$	minimization of the worse $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$0-value over two disjoint subsets
Anisotropic higher Cheeger theory	$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$1	anisotropic perimeter/volume min-max over two disjoint subsets
Two-step graph theory for gaps from $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$2	$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$3	geometric constant associated to $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$4, the second largest eigenvalue of $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$5

Several nearby notions are not “second Cheeger constants” in this sense. The dual Cheeger constant $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$6 is a distinct invariant designed to control the top of the spectrum and near-bipartiteness rather than a higher-order partition problem (Bauer et al., 2012). 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 (Gundert et al., 2014). The average-case $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$7-fold constant of Kenter and Racliffe satisfies $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$8, so it does not single out a separate invariant called “the second Cheeger constant” (Kenter et al., 2015).

2. The two-step graph constant associated with the second largest spectral gap from $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$9

For a finite, simple graph $1$0 with no isolated vertices, the normalized Laplacian in the 2026 formulation is

$1$1

with

$1$2

Because $1$3 is always the trivial largest distance from $1$4, the quantity of interest is

$1$5

which the paper calls the “second largest spectral gap from $1$6.” A basic structural identity is

$1$7

Thus $1$8 simultaneously reflects poor expansion at the bottom of the spectrum and near-bipartiteness at the top (Beers et al., 6 Jun 2026).

The spectral backbone is the two-step operator

$1$9

whose eigenvalues are $\tilde h$0. With the weighted inner product

$\tilde h$1

the Rayleigh quotient is

$\tilde h$2

Since the largest eigenvalue of $\tilde h$3 is $\tilde h$4, corresponding to constants, $\tilde h$5 is the second largest eigenvalue of $\tilde h$6, and

$\tilde h$7

The corresponding geometric quantity is

$\tilde h$8

where

$\tilde h$9

For indicator functions,

$h_k(\Omega)$0

This identity is the direct analogue of the usual role of set indicators in classical Cheeger theory, but for the two-step operator $h_k(\Omega)$1. The same paper notes that this $h_k(\Omega)$2 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 (Beers et al., 6 Jun 2026).

3. Relation to classical and dual Cheeger constants

The classical Cheeger constant and the dual Cheeger constant are

$h_k(\Omega)$3

and

$h_k(\Omega)$4

They govern the bottom and top spectral edges through

$h_k(\Omega)$5

Since

$h_k(\Omega)$6

the 2026 paper first derives a combined bound from

$h_k(\Omega)$7

namely

$h_k(\Omega)$8

This shows that $h_k(\Omega)$9 cannot be described naturally by $k=2$0 alone or by $k=2$1 alone (Beers et al., 6 Jun 2026).

The interpretation of $k=2$2 is genuinely two-step. For

$k=2$3

$k=2$4 is the probability that a one-step random walk started from $k=2$5 leaves $k=2$6, so $k=2$7 is the probability that it stays in $k=2$8 after one step. By contrast,

$k=2$9

is the probability that a walk started in $h_{2,F}$0 returns to $h_{2,F}$1 after two steps, with the starting vertex in $h_{2,F}$2 sampled proportional to degree. The key point is that $h_{2,F}$3 detects one-step escape from a bottleneck, whereas $h_{2,F}$4 detects two-step return behavior (Beers et al., 6 Jun 2026).

The main sharp Cheeger-type inequalities are

$h_{2,F}$5

The lower bound

$h_{2,F}$6

is the new inequality involving $h_{2,F}$7; the upper bound is sharp with equality if and only if $h_{2,F}$8 is bipartite or disconnected. The same paper also proves

$h_{2,F}$9

with equality if and only if $1$0 is bipartite or disconnected, and

$1$1

These facts place $1$2 between ordinary expansion and near-bipartite two-step return structure (Beers et al., 6 Jun 2026).

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

$1$3

shows that $1$4 is a complementary invariant rather than a two-cluster one (Bauer et al., 2012).

4. The second Cheeger constant in the higher Cheeger problem

In the continuum higher Cheeger problem of Bobkov and Parini, the $1$5-th Cheeger constant of a measurable set $1$6 is

$1$7

The specialization $1$8 gives the second Cheeger constant

$1$9

Equivalently,

$\tilde h$0

A minimizing pair is a Cheeger 2-tuple, or a pair of coupled Cheeger sets (Bobkov et al., 2017).

For bounded measurable $\tilde h$1, existence holds in a strong adjusted form: there exists a $\tilde h$2-adjusted Cheeger couple. The central equilibrium condition is the $\tilde h$3-adjusted relation

$\tilde h$4

This shows that each component is itself a Cheeger set of the complement of the other. The theory does not require

$\tilde h$5

so a residual set $\tilde h$6 may remain.

For bounded open $\tilde h$7, regularity of $\tilde h$8-adjusted tuples is substantial. Each $\tilde h$9 is $h_2(\Omega)$0 for every $h_2(\Omega)$1, the singular set has Hausdorff dimension at most $h_2(\Omega)$2, and if $h_2(\Omega)$3 then $h_2(\Omega)$4 is $h_2(\Omega)$5. The reduced free boundary has constant mean curvature

$h_2(\Omega)$6

while for a $h_2(\Omega)$7-adjusted tuple the interface $h_2(\Omega)$8 has constant mean curvature. If $h_2(\Omega)$9 denotes the interface curvature measured from inside $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$00 and $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$01, then

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$02

and if $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$03, then

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$04

These curvature relations are the geometric balance laws of the second Cheeger problem (Bobkov et al., 2017).

The spectral link is exact for the second $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$05-Laplacian eigenvalue. If

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$06

then

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$07

More generally,

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$08

For $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$09, the second Cheeger constant is therefore the $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$10 limit of the second Dirichlet eigenvalue.

The planar theory includes explicit model domains. For a disk $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$11,

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$12

and the minimizing pair consists of the Cheeger sets of two half-disks. For the annulus

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$13

one has

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$14

where $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$15 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 (Bobkov et al., 2017).

5. The anisotropic second Cheeger constant

In the anisotropic setting, one fixes an even, convex, positively $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$16-homogeneous $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$17 integrand $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$18, uniformly equivalent to the Euclidean norm, and defines the anisotropic perimeter

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$19

For Lipschitz boundary,

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$20

The first anisotropic Cheeger constant is

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$21

and the second anisotropic Cheeger constant is

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$22

Equivalently,

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$23

The infimum is attained by connected sets $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$24, called a pair of coupled anisotropic Cheeger sets (Piscitelli, 29 Mar 2025).

The associated spectral operator is the anisotropic $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$25-Laplacian

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$26

Its second Dirichlet eigenvalue satisfies a domain decomposition formula: $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$27 This is the precise spectral counterpart of the min-max definition of $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$28. The paper proves the anisotropic second Cheeger inequality

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$29

and the exact asymptotic identification

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$30

A key geometric feature is the nodal-domain mechanism. If $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$31 is a second eigenfunction with nodal domains $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$32 and $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$33, then

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$34

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 $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$35 one has only

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$36

whereas for $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$37 the nodal decomposition aligns exactly with the two-set Cheeger problem.

The anisotropic theory also includes comparison principles. One has

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$38

with strict inequality when $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$39 has a unique anisotropic Cheeger set, and an anisotropic Hong–Krahn–Szegő bound

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$40

where $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$41 is the union of two disjoint Wulff shapes, each of measure $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$42. The paper further proves existence of a $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$43-adjusted anisotropic Cheeger couple satisfying

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$44

This is the anisotropic analogue of the adjustment structure in the Euclidean higher Cheeger problem (Piscitelli, 29 Mar 2025).

6. Distinct but adjacent notions

The dual Cheeger constant on graphs is not a second Cheeger constant. In the infinite-graph theory,

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$45

and it is designed to capture closeness to bipartiteness and to control the top eigenvalue through inequalities such as

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$46

The paper introducing $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$47 explicitly states that it is not a second or higher-order Cheeger constant (Bauer et al., 2012).

Higher-dimensional discrete Cheeger theory for simplicial complexes is also different. Parzanchevski–Rosenthal–Tessler’s combinatorial Cheeger constant $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$48 and the later extension to arbitrary complexes concern the higher-dimensional analogue of the graph spectral gap: $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$49 with

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$50

This is a higher-dimensional analogue of the graph’s second smallest Laplacian eigenvalue side, not a second Cheeger constant in the multiway sense (Gundert et al., 2014).

The $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$51-fold constant of Kenter and Racliffe is an average-case multiway invariant,

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$52

and it satisfies

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$53

Accordingly, its $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$54 specialization is exactly the ordinary Cheeger constant, not a separate “second” constant (Kenter et al., 2015).

Finally, sheaf-theoretic coboundary expansion on graphs generalizes the ordinary Cheeger constant via

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$55

but that theory does not define or analyze a second Cheeger constant, a multiway partition constant, or any two-cluster analogue (First et al., 2022).

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 $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$56, and in the anisotropic/Finsler setting to $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$57. In the recent graph theory of the second largest spectral gap from $$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$58, the natural analogue is instead the two-step return constant

$$h_{2,F}(\Omega)=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\}: E_{1},E_{2}\subset \Omega,\ E_{1}\cap E_{2}=\emptyset,\ |E_1|,|E_2|>0\right\}.$$59

which is a new Cheeger-type quantity rather than the standard higher-order one (Bobkov et al., 2017, Piscitelli, 29 Mar 2025, Beers et al., 6 Jun 2026).