As in the previous section, let (*M, g,*m) be a weighted Riemannian manifold with Ric_{∞} *≥*1
and m(*M*) = 1, fix *θ* *∈* (0*,*1) and take a Borel set *A* *⊂* *M* with m(*A*) = *θ*. We employ
the needle decomposition associated with *f* := *χ*_{A} *−θ* as in Subsection 2.3: (*Q, ν*),
*{*(*X*_{q}*,*m_{q})*}**q∈Q*, and the guiding function *u* with ∫

*M**u d*m= 0. Set *A*_{q}:=*A∩X*_{q} as in the
previous section.

Put *δ*(*A*) :=P(*A*)*− I*(R*, γ*)(

*θ*) and define

*Q*

_{ℓ}:={

*q* *∈Q*m_{q}(*A*_{q}) =*θ,* P(*A*_{q})*− I*(R*, γ*)(

*θ*)

*<*√

*δ*(

*A*)}

(7.1)
as a set of ‘long’ needles (recall from Lemma 2.6 that small deficit implies large diame-
ter). Notice that *Q*_{ℓ} is a measurable set since the function *q* *7→* P(*A*_{q}) is measurable by
[CMM, Lemma 4.1]. We observe from Lemma 6.1 the following (similarly to the proof of
Theorem 6.2).

**Lemma 7.1 (***Q*_{ℓ} **is large)** *We have* *ν*(*Q*_{ℓ})*≥*1*−*√
*δ*(*A*)*.*

For further analyzing the behavior of long needles, we define
*Q*^{−}_{ℓ} :={

*q* *∈Q*_{ℓ}m*q*

(*A*_{q}*△*(*−∞, r*^{−}_{m}

*q*(*θ*)])

*≤*√
*δ*(*A*)}

*,*
*Q*^{+}_{ℓ} :={

*q* *∈Q*_{ℓ}m_{q}(

*A*_{q}*△*[*r*^{+}_{m}_{q}(*θ*)*,∞*))

*≤*√
*δ*(*A*)}

*,*

(7.2)
where *X*_{q} is parametrized by*u* and *r*_{m}^{±}_{q}(*θ*)*∈X*_{q} are defined by

m_{q}(

*X*_{q}*∩*(*−∞, r*^{−}_{m}_{q}(*θ*)])

=m_{q}(

*X*_{q}*∩*[*r*^{+}_{m}_{q}(*θ*)*,∞*))

=*θ*

as in Proposition 4.1. The measurability of *Q*^{+}_{ℓ} and *Q*^{−}_{ℓ} can be shown as in [CMM] (see
Lemma 6.1 and the paragraph following it). Then the next lemma is a consequence of
Proposition4.1.

**Lemma 7.2 (***Q*^{−}_{ℓ} *∪Q*^{+}_{ℓ} **is large)** *If* *δ*(*A*) *is suﬃciently small, then we have*
*ν*(

*Q**ℓ**\*(*Q*^{−}_{ℓ} *∪Q*^{+}_{ℓ} ))

*≤*√
*δ*(*A*)*.*

*Proof.* Recall from Theorem 2.10 that, for *ν*-almost every *q* *∈* *Q*, (*X*_{q}*,*m_{q}) satisfies
Ric_{∞}*≥*1 and m_{q}(*A*_{q}) = *θ*. Then we deduce from (4.1) and lim_{δ}_{→}_{0}*C*_{5}(*θ, δ*) = *∞* that

P(*A*_{q})*− I*(*X**q**,*m*q*)(*θ*)*≥*min
{

m_{q}(

*A*_{q}*△*(*−∞, r*_{m}^{−}_{q}(*θ*)])
*,*m_{q}(

*A*_{q}*△*[*r*_{m}^{+}_{q}(*θ*)*,∞*))}

for *q∈Q*_{ℓ} provided that *δ*(*A*) is suﬃciently small. Hence

P(*A*_{q})*− I*(R*, γ*)(

*θ*)

*≥*P(

*A*

_{q})

*− I*(

*X*

*q*

*,*m

*q*)(

*θ*)

*>*√

*δ*(

*A*) for

*q∈Q*

*ℓ*

*\*(

*Q*

^{−}

_{ℓ}

*∪Q*

^{+}

_{ℓ}), and it follows from Lemma 6.1 that

*δ*(*A*)*≥*√

*δ*(*A*)*·ν*(

*Q*_{ℓ}*\*(*Q*^{−}_{ℓ} *∪Q*^{+}_{ℓ}))
*.*

*2*

Next we shall show that one of *Q*^{−}_{ℓ} and *Q*^{+}_{ℓ} necessarily has a small volume. This is
the most technical step in this section and the structure of the proof diﬀers from that
of [CMM, Proposition 6.4], due to the fact that the diameter of *M* is not bounded and
needles can be infinitely long (cf., for example, [CMM, Proposition 5.1, Corollary 5.4]).

The following observation by virtue of (6.2) will play a crucial role. Recall that *a*_{θ} *∈*R is
defined by * γ*((

*−∞, a*

*θ*]) =

*θ*.

**Proposition 7.3 (***u* **is nearly centered on most needles)** *Ifδ*(*A*)*is suﬃciently small,*
*then there exists a measurable set* *Q*_{c}*⊂Q* *such that* *ν*(*Q*_{c})*≥*1*−δ*(*A*)(1*−ε*)*/*(9*−*3*ε*) *and*

max{

*|a*_{θ}*−r*^{−}_{m}

*q*(*θ*)*|,|a*_{1}_{−}_{θ}*−r*_{m}^{+}

*q*(*θ*)*|*}

*≤C*_{8}(*θ, ε*)*δ*(*A*)^{(1}^{−}^{ε)/(9}^{−}^{3ε)} (7.3)
*for every* *q* *∈Q*_{c}*∩Q*_{ℓ}*.*

*Proof.* We set*δ* :=*δ*(*A*) and

*a*:= 2(1*−ε*)
3(3*−ε*)

for simplicity, and observe from (6.2) that the set*Q*_{c}*⊂Q* consisting of *q* with
( ∫

*X**q*

*u d*m*q*

)2

*≤C*7(*θ, ε*)*δ*^{a} (7.4)

satisfies *ν*(*Q*_{c}) *≥* 1 *−δ*(1*−ε*)*/*(3*−ε*)*−a*. Fix a needle *q* *∈* *Q*_{c} *∩* *Q*_{ℓ} and put m_{q} = e^{−ψ}*dx*,
*r*^{−} :=*r*_{m}^{−}_{q}(*θ*) and *r*^{+} :=*r*_{m}^{+}_{q}(*θ*) for brevity.

Since the assertion is symmetric, by reversing*X*_{q}if necessary, we can assume*I*(*X**q**,*m*q*)(*θ*) =
e^{−}^{ψ(r}^{−}^{)}. Then we have e^{−}^{ψ(r}^{−}^{)} *≤*P(*A*_{q})*≤ I*(R*, γ*)(

*θ*) +

*√*

*δ* and deduce from (3.3) that
*ψ*(*x*)*− ψ*

_{g}(

(*x−r*^{−}) +*a*_{θ})

*≥*(

*ψ*_{+}^{′} (*r*^{−})*−a*_{θ})

(*x−r*^{−})*−ω*(*θ*)*√*
*δ*

on *X*_{q}, where we recall that *X*_{q} is parametrized by *u*. We similarly observe from (3.4)
that

*ψ*(*x*)*− ψ*

_{g}(

(*x−r*^{−}) +*a*_{θ})

*≤*(

*ψ*^{′}_{+}(*r*^{−})*−a*_{θ})

(*x−r*^{−}) +*ω*(*θ*)*δ*^{1/4}

on [*S* +*r*^{−}*−a**θ**, T* +*r*^{−}*−a**θ*]. Let us set *α* := *a**θ* *−r*^{−}, *β* := *ψ*^{′}_{+}(*r*^{−})*−a**θ* and observe

*|β| ≤* (*C*_{2}+ 1)*√*

*δ* from (3.9). By (7.4) we also find that *α* *→*0 as *δ* *→* 0, our goal is to
make this quantitative.

We have

∫

*X**q*

*u d*m_{q} =

∫

*X**q*

*x*m_{q}(*dx*)

*≤*

∫ _{∞}

0

*x*exp

(*− ψ*

_{g}(

*x*+

*α*)

*−β*(

*x−r*

^{−}) +

*ω√*

*δ*

)
*dx*

+

∫ _{0}

*S**−**α*

*x*exp

(*− ψ*

_{g}(

*x*+

*α*)

*−β*(

*x−r*

^{−})

*−ωδ*

^{1/4})

*dx*

= 1

*√*2*π*

∫ _{∞}

0

*x*exp
(

*−*(*x*+*α*+*β*)^{2}

2 +*αβ*+ *β*^{2}

2 +*βr*^{−}+*ω√*
*δ*

)
*dx*

+ 1

*√*2*π*

∫ 0
*S**−**α*

*x*exp
(

*−*(*x*+*α*+*β*)^{2}

2 +*αβ* +*β*^{2}

2 +*βr*^{−}*−ωδ*^{1/4}
)

*dx*

= exp (

*αβ*+*β*^{2}

2 +*βr*^{−}+*ω√*
*δ*

) ∫ _{∞}

*α*+*β*

(*x−α−β*)* γ*(

*dx*) + exp

(

*αβ*+ *β*^{2}

2 +*βr*^{−}*−ωδ*^{1/4}

) ∫ *α*+*β*
*S*+*β*

(*x−α−β*)* γ*(

*dx*)

*.*

Since *|β| ≤*(*C*_{2}+ 1)*√*

*δ* and *α→*0 as *δ* *→*0, we find
exp

(

*αβ*+ *β*^{2}

2 +*βr*^{−}+*ω√*
*δ*

)

*≤*1 +*C*(*θ*)*√*
*δ,*

exp (

*αβ*+*β*^{2}

2 +*βr*^{−}*−ωδ*^{1/4}
)

*≥*1*−C*(*θ*)*δ*^{1/4}*.*
Moreover, we observe

∫ _{α+β}

*S*+*β*

(*x−α−β*)* γ*(

*dx*) =

∫ _{α+β}

*−∞*

(*x−α−β*)* γ*(

*dx*)

*−*

∫ _{S+β}

*−∞*

(*x−α−β*)* γ*(

*dx*)

=

∫ *α*+*β*

*−∞*

(*x−α−β*)* γ*(

*dx*) + 1

*√*2*π*

[e^{−}^{x}^{2}^{/2}]*S*+*β*

*−∞* + (*α*+*β*)* γ*(

(*−∞, S*+*β*])
and, assuming that *δ* is suﬃciently small,

e^{−}^{(S+β)}^{2}^{/2} *≤*e^{−}^{(1}^{−}^{ε)S}^{2}^{/2} = (1*−S*)^{(1}^{−}^{ε)}^{2}e^{−}^{ε(1}^{−}^{ε)S}^{2}^{/2}

(e^{−}^{S}^{2}^{/2}
1*−S*

)_{(1−ε)}^{2}

*≤C*(*θ, ε*)*δ*^{(1}^{−}^{ε)}^{2}^{/4}
and

* γ*(

(*−∞, S*+*β*])

*≤ γ*(

(*−∞, S*])

+*√|β|*

2*π* *≤C*(*θ*)*δ*^{1/4}
by (3.19) and (7.1). Therefore we obtain

∫

*X**q*

*u d*m_{q}*≤*

∫ _{∞}

*−∞*

(*x−α−β*)* γ*(

*dx*) +

*C*(

*θ, ε*)

*δ*

^{(1}

^{−}

^{ε)}

^{2}

^{/4}

=*−α−β*+*C*(*θ, ε*)*δ*^{(1}^{−}^{ε)}^{2}^{/4}

*≤ −α*+*C*(*θ, ε*)*δ*^{(1−ε)}^{2}^{/4}*.*

A similar calculation shows

∫

*X**q*

*u d*m_{q} *≥ −α−C*(*θ, ε*)*δ*^{(1}^{−}^{ε)}^{2}^{/4}

as well. Combining these with (7.4) yields (provided that *a/*2*≤*(1*−ε*)^{2}*/*4)

*|α| ≤*
*α*+

∫

*X**q*

*u d*m_{q}
+

∫

*X**q*

*u d*m_{q}

*≤C*(*θ, ε*)*δ*^{a/2}*.* (7.5)

In order to bound *|a*_{1}_{−}_{θ}*−r*^{+}*|*, let us recall
e^{−}^{ψ(x)}*≤* 1

*√*2*π*exp
(

*αβ* +*β*^{2}

2 +*βr*^{−}+*ω√*
*δ*

) exp

(

*−*(*x*+*α*+*β*)^{2}
2

)

*≤*(

1 +*C*(*θ*)*√*
*δ*)

e^{−}^{ψ}^{g}^{(x+α+β)}

on*X*_{q}. Therefore, on one hand, for Θ*>−*(*α*+*β*) with e^{−}^{ψ}^{g}^{(a}^{1−θ}^{+Θ+α+β)} *≥*e^{−}^{ψ}^{g}^{(a}^{1−θ}^{)}*/*2,
m_{q}(

[*a*_{1}_{−}_{θ}+ Θ*,∞*))

*≤*(

1 +*C*(*θ*)*√*
*δ*)

* γ*(

[*a*_{1}_{−}_{θ}+ Θ +*α*+*β,∞*))

*≤*(

1 +*C*(*θ*)*√*
*δ*)(

*θ−* e^{−ψ}^{g}^{(a}^{1−θ}^{)}

2 (Θ +*α*+*β*)
)

*.*

Then choosing

Θ = 2e^{ψ}^{g}^{(a}^{1−θ}^{)}*θC*(*θ*)*√*

*δ−α−β*
implies m_{q}([*a*_{1}_{−}_{θ}+ Θ*,∞*))*< θ* and hence

*r*^{+} *< a*1*−**θ*+ Θ*≤a*1*−**θ*+*C*(*θ, ε*)*δ*^{a/2}*,*

where we used (7.5). On the other hand, for Ξ *> α* + *β* with e^{−}^{ψ}^{g}^{(a}^{1}^{−}^{θ}^{−}^{Ξ+α+β)} *≥*
e^{−}^{ψ}^{g}^{(a}^{1−θ}^{)}*/*2, we observe

m*q*

([*a*1*−**θ**−*Ξ*,∞*))

*≥*1*−*(

1 +*C*(*θ*)*√*
*δ*)

* γ*(

(*−∞, a*1*−**θ**−*Ξ +*α*+*β*])

*≥*1*−*(

1 +*C*(*θ*)*√*
*δ*)(

(1*−θ*)*−* e^{−}^{ψ}^{g}^{(a}^{1}^{−}^{θ}^{)}

2 (Ξ*−α−β*)
)

*.*

This yieldsm_{q}([*a*_{1}_{−}_{θ}*−*Ξ*,∞*))*> θ* with Ξ = 2e^{ψ}^{g}^{(a}^{1}^{−}^{θ}^{)}(1*−θ*)*C*(*θ*)*√*

*δ*+*α*+*β*, and hence
*r*^{+} *> a*_{1}_{−}_{θ}*−*Ξ*≥a*_{1}_{−}_{θ}*−C*(*θ, ε*)*δ*^{a/2}*.*

This completes the proof. *2*

Let us explain the geometric intuition of the proof of the next proposition. If both
*ν*(*Q*^{−}_{ℓ}) and*ν*(*Q*^{+}_{ℓ} ) have a certain volume, then the strict concavity of*I*(R*, γ*)implies that the
sum of the perimeters of regions

*A*

^{−}and

*A*

^{+}corresponding to

*Q*

^{−}

_{ℓ}and

*Q*

^{+}

_{ℓ}, respectively, is larger than

*I*(R

*,*)(

**γ***θ*). This contradicts the assumed small deficit when the gap between P(

*A*) and P(

*A*

^{−}) +P(

*A*

^{+}) is suﬃciently small. In order to construct such a decomposition

*−* *̸*

**Proposition 7.4 (One of** *Q*^{−}_{ℓ} **and** *Q*^{+}_{ℓ} **is small)** *Assume* *θ* *̸*= 1*/*2*. Then we have*
min*{ν*(*Q*^{−}_{ℓ})*, ν*(*Q*^{+}_{ℓ})*} ≤C*_{9}(*θ*)*δ*(*A*)^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*,*

*provided that* *δ*(*A*) *is suﬃciently small.*

*Proof.* Put *δ* = *δ*(*A*) again in this proof. Let us first assume *θ* *∈* (0*,*1*/*2) and consider
the decomposition of *A*,

*A*^{−}_{r} :=*A∩ {u≤r},* *A*^{+}_{r} :=*A∩ {u≥r},*
for *r∈*(*r*_{1}*, r*_{2}) with

*r*_{1} := 2
3*a*_{θ}+1

3*a*_{1}_{−}_{θ}*,* *r*_{2} := 1
3*a*_{θ}+ 2

3*a*_{1}_{−}_{θ}*.*

Note that *a*_{θ} *<* 0 *< a*_{1}_{−}_{θ} = *−a*_{θ} since *θ <* 1*/*2. Moreover, letting *δ* smaller if necessary,
we find from (7.3) that*r*_{1} *≥r*_{m}^{−}_{q}(*θ*) holds for*q* *∈Q*_{c}*∩Q*_{ℓ}.

Since*|∇u|*= 1 almost everywhere, we obtain from the coarea formula (see, e.g., [Cha])
that

m(

*A∩ {r*1 *< u < r*2*}*)

=

∫

*A**∩ {**r*1*<u<r*2*}**|∇u|d*m=

∫ *r*2

*r*1

*|A∩u*^{−}^{1}(*r*)*|dr,*

where *| · |* denotes the (*n−*1)-dimensional measure induced from m (precisely, e^{−}^{Ψ}*H*^{n}^{−}^{1}
where *H*^{n}^{−}^{1} is the (*n−*1)-dimensional Hausdorﬀ measure). For *q∈Q*_{c}*∩Q*^{−}_{ℓ} , we deduce
from*r*_{1} *≥r*_{m}^{−}

*q*(*θ*) and (7.2) that
m_{q}(

*A*_{q}*∩*(*−∞, r*_{1}])

=m_{q}(*A*_{q})*−*m_{q}(

*A*_{q}*\*(*−∞, r*_{1}])

*≥θ−√*

*δ.* (7.6)

Similarly m_{q}(*A*_{q} *∩* [*r*_{2}*,∞*)) *≥* *θ* *−* *√*

*δ* holds for *q* *∈* *Q*_{c} *∩* *Q*^{+}_{ℓ}. Then it follows from
Theorem 2.10(i), Lemmas 7.1,7.2 and Proposition7.3 that

m(*A*^{−}_{r}_{1} *∪A*^{+}_{r}_{2})*≥*

∫

*Q**c**∩**Q*^{−}_{ℓ}

m_{q}(

*A*_{q}*∩*(*−∞, r*_{1}])

*ν*(*dq*) +

∫

*Q**c**∩**Q*^{+}_{ℓ}

m_{q}(

*A*_{q}*∩*[*r*_{2}*,∞*))
*ν*(*dq*)

*≥*(*θ−√*
*δ*)*ν*(

*Q*_{c}*∩*(*Q*^{−}_{ℓ} *∪Q*^{+}_{ℓ} ))

*≥*(*θ−√*

*δ*)(1*−*2*√*

*δ−δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)})

*≥θ−*(1 + 2*θ*)*√*

*δ−θδ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}

*≥θ−δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*.*
Therefore we obtain

∫ *r*2

*r*1

*|A∩u*^{−}^{1}(*r*)*|dr*=*θ−*m(*A*^{−}_{r}_{1} *∪A*^{+}_{r}_{2})*≤δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*,*
and we can choose some ˆ*r∈*(*r*_{1}*, r*_{2}) satisfying

*|A∩u*^{−}^{1}(ˆ*r*)*| ≤* *δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}

*r*_{2}*−r*_{1} = 3*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}

*a*_{1}_{−}_{θ}*−a*_{θ} = 3*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}
2*|a*_{θ}*|* *.*

This yields that

P(*A*^{−}_{r}_{ˆ}) +P(*A*^{+}_{r}_{ˆ})*−*P(*A*)*≤*2*|A∩u*^{−}^{1}(ˆ*r*)*| ≤* 3*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}

*|a*_{θ}*|* *.* (7.7)

In the first inequality, take a sequence *{ϕ*_{i}*}**i**∈*N of Lipschitz functions such that 0 *≤ϕ*_{i} *≤*
*χ*_{A}, *ϕ*_{i} *→* *χ*_{A} in *L*^{1}(m) and lim_{i→∞}∫

*M**|∇ϕ*_{i}*|d*m= P(*A*) (recall (2.2) for the definition of
P(*A*)), and put

*ρ*^{+}_{i} (*x*) := min{

*i·*max*{u*(*x*)*−r,*ˆ 0*},*1}

*,* *ρ*^{−}_{i} (*x*) := 1*−ρ*^{+}_{i} (*x*)*.*

Then *ρ*^{±}_{i} *ϕ*_{i} *→χ*_{A}*±*
ˆ

*r* in *L*^{1}(m) and
P(*A*^{−}_{ˆ}_{r}) +P(*A*^{+}_{ˆ}_{r})*≤*lim inf

*i→∞*

∫

*M*

(*|∇*(*ρ*^{−}_{i} *ϕ*_{i})*|*+*|∇*(*ρ*^{+}_{i} *ϕ*_{i})*|*)
*d*m

*≤* lim

*i**→∞*

∫

*M*

(*ρ*^{−}_{i} +*ρ*^{+}_{i} )*|∇ϕ*_{i}*|d*m+ lim inf

*i**→∞*

∫

*M*

(*|∇ρ*^{−}_{i} *|*+*|∇ρ*^{+}_{i} *|*)
*ϕ*_{i}*d*m

*≤*P(*A*) + lim

*i**→∞*

∫

*A**∩{*ˆ*r<u<*ˆ*r*+*i*^{−1}*}*

2*i d*m

=P(*A*) + 2*|A∩u*^{−}^{1}(ˆ*r*)*|.*

Now, it follows from Lemma 2.7 that *I*_{(}^{′′}_{R}_{,γ}_{)} *≤ −I*(R*, γ*)(

*θ*)

^{−}

^{1}on (0

*, θ*] (since

*θ <*1

*/*2), which implies

P(*A*^{−}_{ˆ}_{r})*≥ I*(R*, γ*)

(m(*A*^{−}_{ˆ}_{r}))

*≥* m(*A*^{−}_{r}_{ˆ})

*θ* *I*(R*, γ*)(

*θ*) + 1 2

*I*(R

*,*)(

**γ***θ*)

(

1*−* m(*A*^{−}_{r}_{ˆ})
*θ*

)m(*A*^{−}_{r}_{ˆ})
*θ* *θ*^{2}*.*
Concerning the second term in the RHS, on one hand, we observe from (7.6) that

m(*A*^{−}_{ˆ}_{r})*≥*(*θ−√*

*δ*)*ν*(*Q**c**∩Q*^{−}_{ℓ})*≥* *θ*

2*ν*(*Q**c**∩Q*^{−}_{ℓ})*.*

On the other hand, we similarly find

m(*A*^{−}_{ˆ}_{r}) = *θ−*m(*A*^{+}_{r}_{ˆ})*≤θ−* *θ*

2*ν*(*Q*_{c}*∩Q*^{+}_{ℓ} )*.*

Therefore, setting *V* := min*{ν*(*Q*_{c}*∩Q*^{−}_{ℓ})*, ν*(*Q*_{c}*∩Q*^{+}_{ℓ} )*} ≤*1*/*2, we obtain
P(*A*^{−}_{ˆ}_{r})*≥* m(*A*^{−}_{ˆ}_{r})

*θ* *I*(R*, γ*)(

*θ*) + 1 2

*I*(R

*,*)(

**γ***θ*)

(
1*−* *V*

2
)*V*

2*θ*^{2}*.*

We have a similar inequality for *A*^{+}_{r}_{ˆ} in the same way. Summing up, we obtain
P(*A*^{−}_{ˆ}_{r}) +P(*A*^{+}_{ˆ}_{r})*≥ I*(R*, γ*)(

*θ*) + 1

*I*(R*, γ*)(

*θ*) (

1*−* *V*
2

)*V*

2*θ*^{2} *≥ I*(R*, γ*)(

*θ*) +

*c*(

*θ*)

*V.*

Combining this with (7.7) and *I*(R*, γ*)(

*θ*) =P(

*A*)

*−δ*yields 3

*δ*

^{(1}

^{−}

^{ε)/(9}

^{−}

^{3ε)}

*≥*P(*A*^{−}) +P(*A*^{+})*−*P(*A*)*≥c*(*θ*)*V* *−δ*

and hence, by Proposition 7.3,

min*{ν*(*Q*^{−}_{ℓ})*, ν*(*Q*^{+}_{ℓ} )*} ≤V* +*δ*(1*−ε*)*/*(9*−*3*ε*)

*≤* 1
*c*(*θ*)

(3*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}

*|a*_{θ}*|* +*δ*
)

+*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}

*≤C*(*θ*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*.*
This completes the proof for *θ <*1*/*2.

When *θ >*1*/*2, the complement *A*^{c} of *A* satisfies P(*A*^{c}) =P(*A*) and m(*A*^{c}) = 1*−θ <*

1*/*2. Note also that*I*(R*, γ*)(

*θ*) =

*I*(R

*,*)(1

**γ***−θ*) and

*r*

^{−}

_{m}

_{q}(

*θ*) =

*r*

^{+}

_{m}

_{q}(1

*−θ*),

*r*

_{m}

^{+}

_{q}(

*θ*) =

*r*

_{m}

^{−}

_{q}(1

*−θ*).

Hence we have, since*E\F* =*E∩F*^{c}=*F*^{c}*\E*^{c},

*A*_{q}*△*(*−∞, r*_{m}^{−}_{q}(*θ*)] =*A*^{c}_{q}*△*(*r*^{−}_{m}_{q}(*θ*)*,∞*) =*A*^{c}_{q}*△*(*r*^{+}_{m}_{q}(1*−θ*)*,∞*)

and similarly *A*_{q}*△*[*r*_{m}^{+}_{q}(*θ*)*,∞*) = *A*^{c}_{q}*△*(*−∞, r*^{−}_{m}_{q}(1*−θ*)). Therefore we can obtain the

claim for *A* by applying the above argument to *A*^{c}. *2*

From the proof of Proposition7.4, we find that*C*_{9}(1*−θ*) =*C*_{9}(*θ*) and lim_{θ}_{→}_{1/2}*C*_{9}(*θ*) =

*∞* (since *a*1*/*2 = 0). Hence the case of*θ* = 1*/*2 is not covered.

We finally prove our main theorem. We employ the sub-level and super-level sets of
the guiding function*u* instead of balls in [CMM].

**Theorem 7.5 (Quantitative isoperimetry)** *Let*(*M, g,*m)*be a complete weighted Rie-*
*mannian manifold such that*Ric_{∞}*≥*1*and*m(*M*) = 1*. Fixθ* *∈*(0*,*1)*\{*1*/*2*}andε∈*(0*,*1)*,*
*take a Borel set* *A* *⊂* *M* *with* m(*A*) = *θ, and assume that* P(*A*)*≤ I*(R*, γ*)(

*θ*) +

*δ*

*holds for*

*suﬃciently small*

*δ >*0 (

*relative to*

*θ*

*and*

*ε*)

*. Then, for the guiding function*

*u*

*associated*

*with*

*A*

*such that*∫

*M**u d*m= 0*, we have*
min

{ m(

*A△ {u≤a*_{θ}*}*)
*,*m(

*A△ {u≥a*_{1}_{−}_{θ}*}*)}

*≤C*(*θ, ε*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*.* (7.8)
*Proof.* We set again *δ* = *δ*(*A*). Thanks to Proposition 7.4, we first assume *ν*(*Q*^{+}_{ℓ} ) *≤*
*C*_{9}(*θ*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}. Then we deduce from Lemmas 7.1 and 7.2 that

*ν*(*Q\Q*^{−}_{ℓ}) = *ν*(*Q\Q**ℓ*) +*ν*(

*Q**ℓ**\*(*Q*^{−}_{ℓ} *∪Q*^{+}_{ℓ} ))

+*ν*(*Q*^{+}_{ℓ} )*≤*2*√*

*δ*+*C*9(*θ*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*.*
Therefore we obtain

m(

*A△ {u≤a**θ**}*)

*≤*

∫

*Q*^{−}_{ℓ}

m*q*

(*A**q**△*(*−∞, a**θ*])

*ν*(*dq*) +*ν*(*Q\Q*^{−}_{ℓ} )

*≤*

∫

*Q*^{−}_{ℓ}

m_{q}(

*A*_{q}*△*(*−∞, r*^{−}_{m}_{q}(*θ*)])

*ν*(*dq*) +

∫

*Q*^{−}_{ℓ}

m_{q}(

(*−∞, a*_{θ}]*△*(*−∞, r*_{m}^{−}_{q}(*θ*)])
*ν*(*dq*)

+*ν*(*Q\Q*^{−}_{ℓ} )

*≤*

∫

*Q*^{−}_{ℓ}

m_{q}(

(*−∞, a*_{θ}]*△*(*−∞, r*^{−}_{m}_{q}(*θ*)])

*ν*(*dq*) + 3*√*

*δ*+*C*_{9}(*θ*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*.* (7.9)

In order to estimate the first term, we recall from Proposition 7.3 that *|a*_{θ}*−r*^{−}_{m}_{q}(*θ*)*| ≤*
*C*_{8}(*θ, ε*)*δ*(1*−ε*)*/*(9*−*3*ε*) for *q∈Q*_{c}*∩Q*_{ℓ}. This implies

m*q*

((*−∞, a*_{θ}]*△*(*−∞, r*_{m}^{−}

*q*(*θ*)])

=m*q*

((min*{a*_{θ}*, r*_{m}^{−}

*q*(*θ*)*},*max*{a*_{θ}*, r*_{m}^{−}

*q*(*θ*)*}*])

*≤C*(*θ, ε*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}
for *q∈Q*_{c}*∩Q*_{ℓ}. Substituting this into (7.9), we obtain

m(

*A△ {u≤a*_{θ}*}*)

*≤C*(*θ, ε*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}+*ν*(*Q*^{−}_{ℓ} *\Q*_{c}) + 3*√*

*δ*+*C*_{9}(*θ*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}

*≤C*(*θ, ε*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}*.*

In the case of *ν*(*Q*^{−}_{ℓ} ) *≤* *C*_{9}(*θ*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}, we similarly have m(*A△ {u* *≥* *a*_{1}_{−}_{θ}*}*) *≤*
*C*(*θ, ε*)*δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}. This completes the proof. *2*

We conclude with several remarks and open problems related to Theorem 7.5.

**Remark 7.6** (a) If we assert only the existence of ‘some’ 1-Lipschitz function *u* enjoy-
ing (7.8), then one can merely take *u*(*x*) := *d*(*A, x*) +*a*_{θ}. Therefore the novelty of
Theorem 7.5 lies in the construction of *u* as the guiding function of the needle de-
composition. By construction the guiding function *u* seems closely related to the
Busemann function. When there is a *straight line* *η* : R *−→* *M* (meaning that
*d*(*η*(*s*)*, η*(*t*)) =*|s−t|*for all*s, t∈*R), the associated*Busemann function* **b**:*M* *−→*R
is defined by

**b**(*x*) := lim

*t**→∞*

{*t−d*(

*x, η*(*t*))}

*.*

By construction**b**is 1-Lipschitz and sometimes regarded as ‘a distance function from
infinity’. In Cheeger–Gromoll-type splitting theorems (under Ric*N* *≥* 0, see also
(b) below), we show that **b** is totally geodesic and *M* is split into R *×*Σ, where
*{t} ×* Σ = **b**^{−1}(*t*) and *η*_{x}(*t*) := (*t, x*) is a straight line for every *x* *∈* Σ. This is
a similar phenomenon to the rigidity of the Bakry–Ledoux isoperimetric inequality
(under Ric_{∞} *≥* *K >* 0) in Theorem 2.8, where the guiding function plays a similar
role to the Busemann function (see [Ma2] for details). Going back to our quantitative
investigation, the guiding function *u* shares several properties with the Busemann
function: *u* is 1-Lipschitz, most needles are long in both directions (lim_{δ}_{→}_{0}*S* =*−∞*

and lim_{δ}_{→}_{0}*T* =*∞*in Proposition3.2), and the direction of most needles are the same
(Proposition 7.4). When, for instance, some needle is a straight line, one may relate
the associated Busemann function with the guiding function and obtain (7.8) in terms
of that Busemann function. In this direction, moreover, one could expect an ‘almost
splitting theorem’ as metric measure spaces, namely (*M, g,*m) is close to the product
space (R*,| · |, γ*)

*×Y*in some sense (even when there is no infinite needle). This is an interesting and challenging problem, let us recall that Gromov’s precompactness theorem ([Gr,

*§*5.A]) does not apply under Ric

_{∞}

*≥K >*0.

(b) In comparison with the Cheeger–Gromoll-type splitting theorem under Ric_{∞} *≥* 0 in

assumed in Theorem 7.5. In the splitting theorem we claim that the space splits
oﬀ the real line endowed with the Lebesgue measure, and hence an upper bound of
Ψ is necessary to rule out Gaussian spaces (and hyperbolic spaces with very convex
weight functions). Compare this with the rigidity results under Ric_{∞} *≥* *K >* 0 in
Theorems 2.4,2.8.

(c) Since the needle decomposition is available also for Finsler manifolds by [CM, Oh3], one can prove the analogue of Theorem 7.5 for reversible Finsler manifolds verbatim.

In the non-reversible case, however, the needle decomposition does not provide the sharp isoperimetric inequality and it is unclear if one can generalize Theorem7.5. See [Oh3] for more details on the non-reversible situation, and [Oh4] for a derivation of the sharp Bakry–Ledoux isoperimetric inequality for non-reversible Finsler manifolds.

(d) In Theorem 7.5 we restrict ourselves to weighted Riemannian manifolds since the
needle decomposition is not yet known for metric measure spaces satisfying CD(1*,∞*)
or RCD(1*,∞*). We refer to [AM] for the Bakry–Ledoux isoperimetric inequality on
RCD(1*,∞*)-spaces.

(e) There are two open problems related to Theorem 7.5. The first one is the case of
*θ* = 1*/*2. The condition *θ* *̸*= 1*/*2 was used only in Proposition 7.4, where we showed
that one of *Q*^{−}_{ℓ} and *Q*^{+}_{ℓ} has a small volume. If this step is established in some other
way, then all the other steps of the proof work and we can obtain Theorem 7.5 for
*θ* = 1*/*2.

(f) Another open problem is the optimal order of *δ* in (7.8). Our estimate *δ*^{(1}^{−}^{ε)/(9}^{−}^{3ε)}
seems not optimal at all and, compared with the case of Gaussian spaces (recall
(1.1)), the optimal order is likely*√*

*δ*. We remark that the optimal order is not known
also for CD(*N* *−*1*, N*)-spaces studied in [CMM] (*N* *∈* (1*,∞*)), where they obtained
*δ*^{N/(N}^{2}^{+2N}^{−}^{1)} depending on *N* (recall (1.2)).

(g) Inspired by [DF, CF], we expect that the push-forward measure *u*_{∗}m is close to * γ*
in the Wasserstein distance

*W*

_{1}or

*W*

_{2}over R. We may make use of the Talagrand inequality

*W*2(

*u*

_{∗}m

*,*)

**γ**^{2}

*≤*2 Ent

*(*

**γ***u*

_{∗}m) (recall Subsection 6.3).

*Acknowledgements*. We thank Fabio Cavalletti and Max Fathi for discussions during
the workshop “Geometry and Probability” in Osaka (2019). SO was supported in part
by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 19H01786.