We study the behavior of the second eigenfunction of the anisotropic
p
p
-Laplace operator
-\mathcal Q_{p}u:=-{\rm div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right),
−
Q
p
u
:
=
−
d
i
v
(
F
p
−
1
(
∇
u
)
F
ξ
(
∇
u
)
)
,
as
p \to 1^+
p
→
1
+
, where
F
F
is a suitable smooth norm of
\mathbb{R}^{n}
R
n
. Moreover, for any regular set
\Omega
Ω
, we define the second anisotropic Cheeger constant 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\right\},
h
2
,
F
(
Ω
)
:
=
inf
{
max
{
P
F
(
E
1
)
∣
E
1
∣
,
P
F
(
E
2
)
∣
E
2
∣
}
,
E
1
,
E
2
⊂
Ω
,
E
1
∩
E
2
=
∅
}
,
where
P_{F}(E)
P
F
(
E
)
is the anisotropic perimeter of
E
E
, and study the connection with the second eigenvalue of the anisotropic
p
p
-Laplacian. Finally, we study the twisted anisotropic
q
q
-Cheeger constant with a volume constraint.