Quantitative estimates for the Bakry–Ledoux
isoperimetric inequality. II
Cong Hung MAI∗

Shin-ichi OHTA∗,†

July 27, 2022

Abstract
Concerning quantitative isoperimetry for a weighted Riemannian manifold satisfying Ric∞ ≥ 1, we give an L1 -estimate exhibiting that the push-forward of thereference measure by the guiding function (arising from the needle decomposition)is close to the Gaussian measure. We also show Lp - and W2 -estimates in the 1dimensional case.

1

Introduction

This short article is devoted to several further applications of the detailed estimates in[MO] to quantitative isoperimetry. In [MO], on a weighted Riemannian manifold (M, g, m)(with m = e−Ψ volg ) satisfying m(M ) = 1 and Ric∞ ≥ 1, we investigated the stability of
the Bakry–Ledoux isoperimetric inequality [BL]:()P(A) ≥ I(R,γ) m(A)
(1.1)
for any Borel set A ⊂ M , where P(A) is the perimeter of A, γ(dx) = (2π)−1/2 e−x /2 dx isthe Gaussian measure on R, and I(R,γ) is its isoperimetric profile written as2()
θ = γ (−∞, aθ ] .

e−aθ /2I(R,γ) (θ) = √ ,
2π
2

(1.2)

It is known by [Mo, Theorem 18.7] (see also [Ma, §3]) that equality holds in (1.1) for some
A with θ = m(A) ∈ (0, 1) if and only if (M, g, m) is isometric to the product of (R, | · |, γ)
and a weighted Riemannian manifold (Σ, gΣ , mΣ ) of Ric∞ ≥ 1. Moreover, A is necessarilyof the form (−∞, aθ ] × Σ or [−aθ , ∞) × Σ (so-called a half-space). Then, the stability
result [MO, Theorem 7.5] asserts that, if equality in (1.1) nearly holds, then A is close toa kind of half-space in the sense that the symmetric diﬀerence between them has a small
volume.
∗Department of Mathematics, Osaka University, Osaka 560-0043, Japan (hungmcuet@gmail.com,
s.ohta@math.sci.osaka-u.ac.jp)†
RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Tokyo 103-0027, Japan

1

The proof as well as the formulation of [MO, Theorem 7.5] are based on the needle
decomposition paradigm (also called the localization), which was established by Klartag[Kl] for Riemannian manifolds and has provided a significant contribution specificallyin the study of isoperimetric inequalities (we refer to [CM] for a generalization to metricmeasure spaces satisfying the curvature-dimension condition, and to [CMM] for a stability
result). The half-space we mentioned above is in fact a sub-level or super-level set of theguiding function arising in the needle decomposition (see Section 3 and [MO] for moredetails). The needle decomposition enables us to decompose a global inequality on M into
the corresponding 1-dimensional inequalities on minimal geodesics in M (called needles ortransport rays). Therefore, a more detailed 1-dimensional analysis on needles will furnish
a better estimate on M .
The 1-dimensional analysis in [MO] is concentrated in Proposition 3.2 in it (restatedin Proposition 2.1 below), which gives a very detailed estimate on the diﬀerence from theGaussian measure γ. In this article, as an application of the analysis developed in [MO],we show an L1 -bound between γ and the push-forward measure u∗ m of m by the guiding
function u:∥ρ · eψg − 1∥L1 (γ) ≤ C(θ, ε)δ (1−ε)/(9−3ε) ,where u∗ m = ρ dx and γ = e−ψg dx (see Theorem 3.1 for the precise statement). In the
1-dimensional case (on intervals), we also prove an Lp -bound with the improved (andsharp) order δ 1/p (Proposition 2.2; see Example 2.3 for the sharpness) and an estimateof the L2 -Wasserstein distance W2 (Proposition 2.4). The use of Lp and W2 (instead of
the volume of the symmetric diﬀerence) is inspired by stability results for the Poincaréand log-Sobolev inequalities (e.g., [BF, BGRS, CF, IK, IM]). We refer to Remark 3.2 forsome further related works and open problems.
Acknowledgements. We are grateful to Emanuel Indrei, whose question on the Lp -estimate
led us to write this paper. CHM was supported by Grant-in-Aid for JSPS Fellows
20J11328. SO was supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 19H01786.

2

Quantitative estimates on intervals

We first consider the 1-dimensional case (on intervals) and establish quantitative stability estimates in terms of the Lp -norm and the W2 -distance. The L1 -bound will be
instrumental to study the Riemannian case in the next section.

2.1

An Lp -estimate

Throughout this section, let I ⊂ R be an open interval equipped with a probability
measure m = e−ψ dx such that ψ is 1-convex in the sense that()
1
ψ (1 − t)x + ty ≤ (1 − t)ψ(x) + tψ(y) − (1 − t)t|x − y|22
for all x, y ∈ I and t ∈ (0, 1). This means that (I, | · |, m) satisfies Ric∞ ≥ 1 (or thecurvature-dimension condition CD(1, ∞)), and (1.1) holds. The 1-dimensional isoperimetric inequality is well investigated in convex analysis. An important fact due to Bobkov
2

[Bo, Proposition 2.1] is that an isoperimetric minimizer can be always taken as a halfspace of the form (−∞, a] ∩ I or [b, ∞) ∩ I. Now we restate [MO, Proposition 3.2], whichis the source of all the estimates. Recall that γ = e−ψg dx is the Gaussian measure.
Proposition 2.1 ([MO]) Fix θ ∈ (0, 1) and suppose that()
m (−∞, aθ ] ∩ I = θ
ande−ψ(aθ ) ≤ e−ψg (aθ ) + δ

(2.1)
(2.2)

hold for suﬃciently small δ > 0 (relative to θ). Then we have( ′)ψ(x) − ψg (x) ≥ ψ+(aθ ) − aθ (x − aθ ) − C(θ)δ

(2.3)

for every x ∈ I, and√( ′)ψ(x) − ψg (x) ≤ ψ+(aθ ) − aθ (x − aθ ) + C(θ) δ

(2.4)

′
for every x ∈ [S, T ] ⊂ I such that limδ→0 S = −∞ and limδ→0 T = ∞, where ψ+
denotesthe right derivative of ψ and C(θ) is a positive constant depending only on θ.

The first condition (2.1) means that I is “centered” in comparison with γ which
satisfies γ((−∞, aθ ]) = θ (as in (1.2)). Note also that e−ψ(aθ ) ≥ e−ψg (aθ ) holds by theisoperimetric inequality (1.1) (since P((−∞, aθ ] ∩ I) = e−ψ(aθ ) ), and then (2.2) tells thatthe deficit of (−∞, aθ ] ∩ I in the isoperimetric inequality is less than or equal to δ.
Besides the above proposition, we also need the following estimate in its proof (see[MO, (3.9)]):|ψ ′ (aθ ) − aθ |lim sup +≤ C(θ).
(2.5)
δδ→0The lower bound (2.3) enables us to obtain the following Lp -estimate between γ = e−ψg dx
and m = eψg −ψ γ|I . (We remark that the upper bound (2.4) will not be used.)
Proposition 2.2 (An Lp -estimate on I) Assume (2.1) and (2.2). Then we have∥eψg −ψ − 1∥Lp (γ) ≤ C(p, θ)δ 1/p
for all p ∈ [1, ∞) and suﬃciently small δ > 0 (relative to θ and p), where we set eψg −ψ := 0
on R \ I.
Proof. In this proof, we denote by C a positive constant depending on θ, and put a := aθ
for brevity. Since eψg −ψ − 1 ≥ −1 and m(I) = γ(R) = 1, we find
∫
∫ ∞
[
]p
[ ψg −ψ
]p
p
ψg −ψ
∥e
− 1∥Lp (γ) =1 − eψg −ψ + dγe
− 1 + dγ +
∫I
∫−∞
∞ [
]
[ ψg −ψ]pe− 1 + dγ +1 − eψg −ψ + dγ≤
∫I∫−∞[ ψg −ψ]p[ ψg −ψ]e− 1 + dγ +e
− 1 + dγ,
=
I

I

3

where [r]+ := max{r, 0}. Thus, we need to estimate only [eψg −ψ − 1]+ . Observe that[ (ψg −ψ)(x)
]p
(
)p
e
− 1 + ≤ eCδ|x−a|+Cδ − 1 ≤ ep(Cδ|x−a|+Cδ) − 1
from (2.3) and (2.5), and hence∫∫ ∞[ ψg −ψ]p( p(Cδ|x−a|+Cδ))e− 1 + dγ ≤e
− 1 γ(dx)
I
−∞
)
( 2
∫
epCδ ∞
x=√exp − + pCδ|x − a| dx − 1.
2
2π −∞Dividing the integral into (−∞, a] and [a, ∞), we continue the calculation as
( 2
)
( 2
)
∫ a
∫ ∞
x
x
exp − − pCδ(x − a) dx +exp − + pCδ(x − a) dx
2
2
−∞
a
(
)
∫ a
2
2
(x + pCδ)
(pCδ)
=
exp −
+
+ pCaδ dx
2
2
−∞
(
)
∫ ∞
(x − pCδ)2 (pCδ)2+− pCaδ dx
+
exp −
2
2
a
) {∫ a}((pCδ)2−x2 /2+ pCaδ
edx + pCδ
≤ exp
2
−∞
(
) {∫ ∞}(pCδ)2−x2 /2+ exp− pCaδ
edx + pCδ
2
a
(
)
(√
)
(pCδ)2≤ exp+ pC|a|δ
2π + 2pCδ .
2
Therefore, we obtain()()∫[ ψg −ψ]p(pCδ)22pCδ
e− 1 + dγ ≤ exp pCδ + pC|a|δ +
1+ √
−1
2
2π
I
≤ C(p, θ)δ.
2

This completes the proof.
We remark that, since
{
(
)
}1/p
(
)
(pCδ)2p(Cδ)2
exp pCδ +−1≥ exp Cδ +
− 1,
2
2

the constant C(p, θ) given by the above proof necessarily depends on p. The order δ 1/p in
Proposition 2.2 may be compared with Lp -estimates in [IK] for the log-Sobolev inequality
on Gaussian spaces. One can see that the order δ 1/p is optimal from the following example.
Example 2.3 Let I = (−D, D) and m = (1 + δ) · γ|I , where δ > 0 is given by γ(I) =(1 + δ)−1 . Then, at θ = 1/2, we have a1/2 = 0, m((−∞, 0] ∩ I) = 1/2,δe−ψ(0) − e−ψg (0) = √ ,
2π
and

(
∥e

ψg −ψ

− 1∥Lp (γ) =δδp+1+δ 1+δ
4

(

)1/p
=

1 + δ p−1
1+δ)1/p
δ 1/p .

2.2

A W2 -estimate

From Proposition 2.1, one can also derive an upper bound of the L2 -Wasserstein distance
between m and γ. We refer to [Vi] for the basics of optimal transport theory. What weneed is only the following Talagrand inequality with γ as the base measure (see [Ta], [Vi,Theorem 22.14]):∫2
W2 (m, γ) ≤ 2 Entγ (m) = 2 (ψg − ψ)eψg −ψ dγ,(2.6)
I

where Entγ (m) is the relative entropy of m with respect to γ. We remark that both γ
and m have finite second moment (by the 1-convexity of ψ).
Proposition 2.4 (A W2 -estimate on I) Assume (2.1) and (2.2). Then we have√
W2 (m, γ) ≤ C(θ) δfor suﬃciently small δ > 0 (relative to θ).
Proof. We again denote aθ by a, and C will be a positive constant depending only on θ.
Similarly to the proof of Proposition 2.2, we observe from (2.3) and (2.5) that
∫
∫ ∞
ψg −ψ
(ψg − ψ)edγ ≤(Cδ|x − a| + Cδ)eCδ|x−a|+Cδ γ(dx)
I
−∞
)
( 2
∫
Cδ Cδ ∞x=√ e(|x − a| + 1) exp − + Cδ|x − a| dx
2
2π
−∞
{∫ ∞
( 2
)
}
x
≤ Cδ
|x − a| exp − + Cδ|x − a| dx + C ,2−∞
where we used( 2)x
exp − + Cδ|x − a| dx ≤ C
2
−∞

∫ ∞

from the proof of Proposition 2.2. Then we have
( 2
)
∫ a
x
(a − x) exp − − Cδ(x − a) dx
2
−∞
(
)∫ a
(
)
(Cδ)2
(x + Cδ)2= exp Caδ +(a − x) exp −
dx
2
2
−∞
{
(
)
[
(
)]a }
∫ a
(x + Cδ)2(x + Cδ)2≤ (1 + Cδ) (a + Cδ)exp −dx + exp −
2
2
−∞
−∞
{ ∫ a
(
)}
2
(a + Cδ)2≤ (1 + Cδ) ae−x /2 dx + Cδ + exp −
2
−∞
∫ a
2
2
≤a
e−x /2 dx + e−a /2 + Cδ.
−∞5We similarly find
( 2
)
∫ ∞
x
(x − a) exp − + Cδ(x − a) dx
2
a
(
)∫ ∞
(
)
(Cδ)2
(x − Cδ)2= exp −Caδ +(x − a) exp −
dx
2
2
a
)
[
(
)]∞ }
{
(
∫ ∞
(x − Cδ)2(x − Cδ)2dx − exp −≤ (1 + Cδ) (−a + Cδ)
exp −
2
2
a
a
{ ∫ ∞
(
)}
2
(a − Cδ)2≤ (1 + Cδ) −ae−x /2 dx + Cδ + exp −
2
a
∫ ∞
2
2
≤ −a
e−x /2 dx + e−a /2 + Cδ.
a

Therefore, together with the Talagrand inequality (2.6), we obtain the desired estimate
W22 (m, γ) ≤ Cδ.
2√
We do not know whether the order δ in Proposition 2.4 is optimal. Since Wp (m, γ) ≤W2 (m, γ) for any p ∈ [1, 2) by the Hölder inequality, we have, in particular, a bound of
the L1 -Wasserstein distance:√
W1 (m, γ) ≤ C(θ) δ.
One can alternatively infer this estimate from the Kantorovich–Rubinstein duality (see[Vi]); in fact,∫ ∞√W1 (m, γ) ≤|x − a|·|e(ψg −ψ)(x) − 1| γ(dx) ≤ C(θ) δ.
−∞We also remark that, when we take a detour via the reverse Poincaré inequality in[MO, Proposition 5.1] and the stability result [CF, Theorem 1.2], we arrive at a weakerestimate
W1 (m, γ) ≤ C(θ, ε)δ (1−ε)/4 .
We refer to [CMS, FGS] for stability results for the Poincaré inequality (equivalently, thespectral gap) on CD(N − 1, N )-spaces and RCD(N − 1, N )-spaces with N ∈ (1, ∞).

3

An L1-estimate on weighted Riemannian manifolds

Next, we consider a weighted Riemannian manifold, namely a connected, complete C ∞ Riemannian manifold (M, g) of dimension n ≥ 2 equipped with a probability measurem = e−Ψ volg , where Ψ ∈ C ∞ (M ) and volg is the Riemannian volume measure. Assuming
Ric∞ ≥ 1, we have the Bakry–Ledoux isoperimetric inequality (1.1).
We begin with an outline of the proof of (1.1) via the needle decomposition (see[Kl]). Given a Borel set A ⊂ M with θ = m(A) ∈ (0, 1), we employ the function
f := χA − θ (χA denotes the characteristic functionof A) and an associated 1-Lipschitz∫
function u : M −→ R attaining the maximum of M f ϕ dm among all 1-Lipschitz functions
ϕ. Then, analyzing the behavior of u, one can build a partition {Xq }q∈Q of M consisting6

of (the image of) minimal geodesics (called needles), and Q is endowed with a probability
measure ν. For ν-almost every q ∈ Q, u|Xq has slope 1 (|u(x) − u(y)| = d(x, y) for all
x, y ∈ Xq ) and Xq is equipped with a probability measure mq such that mq (A ∩ Xq ) = θand (Xq , | · |, mq ) satisfies Ric∞ ≥ 1. Moreover, we have)∫∫ (∫h dm =
h dmq ν(dq)
(3.1)
M

Q

Xq

for all h ∈ L1 (m). Then, (1.1) for A is obtained by integrating its 1-dimensional counterparts for A ∩ Xq with respect to ν.
∫
The 1-Lipschitz function u is called the guiding function. We can assume M u dm = 0
without loss of generality, and Xq will be identified with an interval via u (in other words,
Xq is parametrized by u). Denote mq = e−σq dx and µ := u∗ m = ρ dx. Note that supp µ
is an interval and may not be the whole R. Through the parametrization of Xq by u, wededuce from (3.1) that∫e−σq (x) ν(dq),ρ(x) =(3.2)

Q

where we set e−σq (x) := 0 if x ̸∈ Xq .
Theorem 3.1 (An L1 -estimate on M ) Assume Ric∞ ≥ 1 and fix ε ∈ (0, 1). If P(A) ≤I(R,γ) (θ) + δ holds for some Borel set A ⊂ M with θ = m(A) ∈ (0, 1) and suﬃciently smallδ (relative to θ and ε), then u∗ m = ρ dx satisfies∥ρ · eψg − 1∥L1 (γ) ≤ C(θ, ε)δ (1−ε)/(9−3ε) ,∫where u is the guiding function associated with A such that M u dm = 0.
Proof. First of all, by (3.2) and Fubini’s theorem, we have
∫ ∞ ∫
∫
ψg
ψg −σq∥ρ · e − 1∥L1 (γ) =(e− 1) ν(dq) dγ ≤∥eψg −σq − 1∥L1 (γ) ν(dq).
−∞

Q

Q

q
1
We shall estimate ∥eψg −σ√ − 1∥L (γ) by dividing into “good” needles and “bad” needles.
Note that ν(Qℓ ) ≥ 1 − δ holds for√ }{Qℓ := q ∈ Q mq (A ∩ Xq ) = θ, P(A ∩ Xq ) < I(R,γ) (θ) + δ

by [MO, Lemma 7.1], where P(A ∩ Xq ) denotes the perimeter of A ∩ Xq in (Xq , | · |, mq ).
Moreover, it follows from [MO, Proposition 7.3] that there exists a measurable set Qc ⊂ Qsuch that ν(Qc ) ≥ 1 − δ (1−ε)/(9−3ε) and}{max |aθ − rq− |, |a1−θ − rq+ | ≤ C(θ, ε)δ (1−ε)/(9−3ε)for all q ∈ Qc ∩ Qℓ , where mq ((−∞, rq− ] ∩ Xq ) = mq ([rq+ , ∞) ∩ Xq ) = θ (recall thatγ((−∞, aθ ]) = γ([a1−θ , ∞)) = θ).

7

−On the one hand, for q ∈ Qc ∩Qℓ , note that either P(A∩Xq ) ≥ e−σq (rq ) or P(A∩Xq ) ≥+−e−σq (rq ) holds by [Bo, Proposition 2.1] (recall Subsection 2.1). When P(A∩Xq ) ≥ e−σq (rq ) ,we put−γq (dx) = e−ψg,q (x) dx := e−ψg (x+aθ −rq ) dx,
which is a translation of γ satisfying γq ((−∞, rq− ]) = θ. Then, it follows from Proposi√−−tion 2.2 (with e−σq (rq ) ≤ P(A ∩ Xq ) ≤ e−ψg,q (rq ) + δ) and Cavalieri’s principle that∥eψg −σq − 1∥L1 (γ) ≤ ∥eψg,q −σq − 1∥L1 (γq ) + ∥e−ψg,q − e−ψg ∥L1 (dx)√|aθ − rq− |≤ C(θ) δ + 2 √2π(1−ε)/(9−3ε)≤ C(θ, ε)δ
.
+We have the same bound also in the case where P(A ∩ Xq ) ≥ e−σq (rq ) by reversing I in
Proposition 2.2.
On the other hand, for q ∈ Q \ (Qc ∩ Qℓ ), we have the trivial bound∥eψg −σq − 1∥L1 (γ) ≤ ∥eψg −σq ∥L1 (γ) + ∥1∥L1 (γ) = 2.
Therefore, we obtain()∥ρ · eψg − 1∥L1 (γ) ≤ C(θ, ε)δ (1−ε)/(9−3ε) + 2 1 − ν(Qc ∩ Qℓ ) ≤ C(θ, ε)δ (1−ε)/(9−3ε) .
2Note that q ∈ Qc ∩ Qℓ is well-behaved and can be handled by the 1-dimensional
analysis, whereas one has a priori no information of q ∈ Q \ (Qc ∩ Qℓ ). This could
be a common problem for stability estimates via the needle decomposition (see, e.g.,[MO, Theorem 6.2] showing a reverse Poincaré inequality on a manifold from a sharper
estimate on intervals). In particular, it may be diﬃcult to achieve the same order δ as in
the 1-dimensional case (Proposition 2.2) by the needle decomposition. In the Lp -case, it isunclear (to the authors) with what we can replace the trivial bound ∥eψg −σq − 1∥L1 (γ) ≤ 2.
For the Wasserstein distance W2 or W1 , we have the same problem on the control ofq ∈ Q \ (Qc ∩ Qℓ ).
Remark 3.2 (Further related works and open problems) (a) Theorem 3.1 holds
true also for reversible Finsler manifolds by the same proof (see [MO, Remark 7.6(c)]and [Oh1, Oh2]).
(b) As we mentioned in the introduction, our Lp - and W2 -estimates are inspired by thequantitative stability for functional inequalities. We refer to [BGRS, FIL, IK, IM] forthe study of the log-Sobolev inequality on the Gaussian space:∫1
1∥∇f ∥2
Entγ (f γ) ≤ Iγ (f γ) =dγ,
22 Rn fwhere Iγ (f γ) is the Fisher information of a probability measure f γ with respect toγ. They investigated the diﬀerence between γ and f γ, in terms of the additive deficit8δ(f ) = Iγ (f γ)/2 − Entγ (f γ). For instance, W2 -bounds (under certain convexity andconcavity conditions on f ) were given in [BGRS, IM], and L1 - and Lp -bounds can befound in [IK]. In the setting of weighted Riemannian manifolds satisfying Ric∞ ≥ 1(as in Theorem 3.1), we have only the rigidity (see [OT]) and the stability is an open
problem.
(c) We have seen in [MO, §6] that the reverse forms of the Poincaré and log-Sobolev
inequalities can be derived from the isoperimetric deficit. The reverse Poincaré inequality then implies a W1 -estimate for the push-forward by an eigenfunction thanks
to [BF, Theorem 1.3] (see also [FGS]). We also expect a direct W1 - or W2 -estimate
for the push-forward by the guiding function, which remains an open question (see[MO, Remark 7.6(g)]).
(d) Another direction of research is a generalization to negative eﬀective dimension, i.e.,
RicN ≥ K > 0 with N < −1. We have established rigidity in the isoperimetricinequality in [Ma], thereby it is natural to consider quantitative isoperimetry, though
it seems to require longer calculations.

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