ISOPERIMETRY FOR PRODUCT OF PROBABILITY
MEASURES: RECENT RESULTS
CYRIL ROBERTO
Abstract. We present recent results on the isoperimetric problem for
product of probability measures. For distributions with tails between
exponential and Gaussian we state a dimension free result. We sketch
its proof that relies on a functional inequality of Poincaré type and on
a semi-group argument. Also, we give isoperimetric and concentration
results, depending on the dimension, for measures with heavy tails.
1. Introduction.
talenti
The well known classical isoperimetric inequality in Rn (see [59] for a
survey) asserts that among sets of prescribed Lebesgue measure, balls are
those with minimal boundary measure. Mathematically, for any Borel set
A ⊂ Rn ,
1− 1
|∂A|n−1 ≥ nωn |A|n n
where ωn is the measure of the Euclidean ball. The above constant is optimal
since there is equality for balls.
In this note1 we are interested in isoperimetric inequalities for product
probability measures. Even if most of the results are available in more
general settings, we shall focus only on Rn . We start with some notations.
Let µ be a one dimensional probability measure and µn its n-fold product
in Rn . Then, the boundary measure of any Borel set A ⊂ Rn is given by
the following Minkowski content
µn (Ah \ A)
µns (∂A) = lim inf
h→0
h
Date: June 27, 2010.
1991 Mathematics Subject Classification. 60E15,60B11.
Key words and phrases. Isoperimetry, Poincaré type inequality, Concentration of measure phenomenon, heavy tails distributions, Gaussian measure.
This work was supported by the European Research Council through the “Advanced
Grant” PTRELSS 228032 and by GDRE 224 GREFI-MEFI, CNRS-INdAM.
1As one of the participants of the conference “Inhomogeneous Random Systems” held
in Paris on january 27th and 28th 2009, I was asked to write the following article. I
would like to add that it could have equally been written by any one of my collaborators
whether Franck Barthe, Patrick Cattiaux, Nathael Gozlan or Arnaud Guillin. I would
like to acknowledge their invaluable help. I am particularly
in dept to Franck Barthe that
barthe-private
suggested to us part of the presentation given here [10]. Also I would like to thank all the
organizers of the conference for their kind invitation.
1
2
C. ROBERTO
where
Ah = {x ∈ Rn : d(x, A) ≤ h}
is the h-enlargement of A for the Euclidean distance d. Given the isoperimetric profile
Iµn (a) =
inf
A:µn (A)=a
{µns (∂A)}
a ∈ [0, 1],
the isoperimetric inequality we shall deal with reads as
eq:iso
(1)
µns (∂A) ≥ Iµn (µn (A))
A ⊂ Rn .
Having the classical isoperimetric inequality in mind, two natural questions
arise: is it possible to find the isoperimetric profile a 7→ Iµn (a) as a function
of a and of n? Also, is it possible toeq:iso
find the extremal sets in the latter, i.e.
sets for which there is equality in (1)?
Let us already quote that the second question is too hard in general.
Hence we shall focus on the first one.
As a guideline the reader might keep in mind the family of probability
measures on R,
dµα (x) =
e−
1
|x|α
α
2α α Γ(1 + (1/α))
dx
α > 0.
Thanks to the factor 1/α inside the exponential we have µ2 = γ, the standard Gaussian measure, while µ1 = 12 e−|x| dx is the two-sided exponential
distribution. As we shall see there are three different behaviors of the isoperimetric profile, as a function of n, corresponding to three different regimes,
α < 1 (distributions with heavy tails), α ∈ [1, 2] (between exponential and
Gaussian) and α > 2.
This note does not intend to give a complete overview onledoux-saintflour
the isoperimetric
for a detailed
problem. Our aim is much more modest and we refer to [44] barthe-05,ros
account on this topic and some of its applications and to [9, 55] for its
geometrical aspects and itsbcr-05,bcr-06,bcr-07,cggr
link with convex geometry. In fact we merely
would like to collect from [11, 12, 13, 31] some of our recent results on the
isoperimetric inequality for product of probability measures.
This paper is divided into 6 sections. In the first one we study the isoperimetric inequality on the line where extremal sets are known. This will allow
us to give an explicit expression of the isoperimetric profile in dimension 1,
as a function of a. Then, we deal with product measures in Rn . Section 4 is
dedicated to the analytic proof of one of our theorem, while in Section 5 we
make the connection with the concentration of measure phenomenon. We
end this note with few remarks and perspectives.
2. Isoperimetric inequalities in R
We start with the one dimensional case of R for which extremal sets are
known. In turn we shall deduce the isoperimetric profile explicitly. We
ISOPERIMETRY FOR PRODUCT OF PROBABILITY MEASURES
3
distinguish between two cases, log-concave probability measures and distributions with heavy tails.
2.1. Log concave probability measures. For log concave probability
measures Bobkov computed the extremal sets in the isoperimetric inequality
and deduced the isoperimetric profile.
bobkov-96
Theorem 1 (Bobkov [19] Proposition 2.1). Let dµ(x) = ZΦ−1 e−Φ(x) dx, x ∈
R, be a probability measure with Φ : R → R convex and even. Then half-lines
are extremal sets in the isoperimetric inequality.
talagrand-91
The two-sided exponential dµ1 (x) = 21 e−|x| dx has been studied in [58].
Denote by Fµ : t 7→ µ((−∞, t]) the distribution function of µ (µ as the
theorem). Then for t ≤ 0
Z
1 t+h
dµ(x) = ZΦ−1 e−Φ(t) = Fµ′ (t).
µs (∂(−∞, t]) = lim inf
h→0 h h
Hence, by Bobkov’s Theorem, we have
Iµ (Fµ (t)) = Iµ (µ(−∞, t]) = µs (∂(−∞, t]) = Fµ′ (t).
A similar result holds for t ≥ 0. This readily leads to an explicit expression
of Iµ in the log-concave case:
Iµ (a) = Fµ′ ◦ Fµ−1 (a)
a ∈ [0, 1].
Note that by symmetry Iµ (a) = Iµ (1 − a). Bobkov also proved that Iµ is
concave and that the
concavity of Iµ is actually equivalent to the fact that
bobkov-96
Φ is convex (see [19, Appendix]). Moreover, for any continuous concave
function I : [0, 1] → R+ such that I(0) = 0 and I(1 − a) = I(a), there exists
bobkov-houdre-97
a symmetric log-concave probability measure µ on R for which Iµ = I [27].
2.2. Heavy tails probability measures. For heavy tails probability measures Bobkov and Houdré computed the extremal sets in the isoperimetric
inequality. As for the log-concave case we shall deduce the isoperimetric
profile from their result.
bobkov-houdre-97
Theorem 2 (Bobkov-Houdré [27] Corollary 13.10). Consider the probability
measure dµ(x) = ZΦ−1 e−Φ(x) dx, x ∈ R, with Φ : R → R even. Then extremal
sets in the isoperimetric inequality can be found among half-lines, symmetric
segments and their complements.
The same type of computation as for the log-concave case lead to
′
−1
′
−1 min(a, 1 − a)
.
Iµ (a) = min Fµ ◦ Fµ (a), 2Fµ ◦ Fµ
2
Again by symmetry Iµ (a) = Iµ (1 − a).
4
C. ROBERTO
2.3. Examples. Computations are easy for the two-sided exponential measure dµ1 (x) = 21 e−|x| dx. One has
Iµ1 (a) = min(a, 1 − a).
|x|α
For the exponential type distribution dµα (x) =
e− α
dx
1
α
2α Γ(1+(1/α))
with α 6=
1 the computation is not explicit, but the following asymptotic holds
lim
a→0
Iµα (a)
1− 1 = 1
α
a α log a1
barthe-01,bobkov-zegarlinski
and similarly when a → 1 (see [7, 29]). In particular, for any α > 0, there
exists two constants c, c′ > 0 (that might depend on α) such that
1
1
1 1− α
1 1− α
′
≤ Iµα (a) ≤ c p log
with p = min(a, 1 − a).
cp log
p
p
For Cauchy-type distributions of the form dmα (x) =
1
1+ α
−1 (a) = α21/α min(a, 1 − a)
′
◦ Fm
α > 0 one gets Fm
α
α
α
dx
2(1+|x|)1+α
with
and thus,
1
Imα (a) = α min(a, 1 − a)1+ α .
Similar computations can be done for the more general probability mea2
sure dµΦ (x) = ZΦ−1 e−Φ(x)
√dx. Assume that Φ is C and even. If in addition
either Φ is convex and Φ is concave on (0, ∞) or Φ is concave on (0, ∞)
satisfying Φ(x)/x
→ 0 and Φθ convex for some θ > 1, then IµΦ (a) compares
bcr-07
to pΦ′ ◦ Φ−1 log p1 with p = min(a, 1 − a). See [13, Proposition 13] and
cggr
[31, Proposition 3.18] for more details and the proofs.
3. Isoperimetric inequalities in Rn
We start with a simple observation. Given a Borel set A ⊂ Rn , we have
(∂A ×
on one hand µn+1 (A × R) = µn (A), and on the other hand µn+1
s
R) = µns (∂A). Hence, by the very definition of the isoperimetric profile,
Iµn+1 ≤ Iµn . In particular
Iµ ≥ Iµ2 ≥ · · · ≥ Iµn ≥ · · · ≥ Iµ∞ := inf Iµn .
n≥1
Quite remarkable is the fact that for the Gaussian measure µ2 there is equality in the previous sequence. This is a direct consequence of the following
theorem.
sudakov-tsirelson
Borell-75
Theorem 3 (Sudakov-Tsirel’son [57], Borell [30]). For any dimension n,
half-spaces are extremal sets in the isoperimetric problem for the standard
Gaussian measure µn2 . In other words Iµ2 = Iµn2 for any n.
ISOPERIMETRY FOR PRODUCT OF PROBABILITY MEASURES
5
The fact that Iµ∞ = Iµ is actually characteristic of the Gaussian measures.
Indeed, it can be shown that Gaussian measures are the only symmetric
measures on the line such that for any dimension n the coordinate halfspaces {x ∈ Rn : x1 ≤ t} are extremal sets inbobkov-houdre-96bis,kwapien,oleszkiewicz
the isoperimetric inequality
for the corresponding product measure. See [28, 41, 54] for more precise
(and stronger) statements.
An alternative
proof of the latter Gaussian isoperimetric result was given
bobkov-97
by Bobkov [21]. He proved the following functional inequality
Z
Z q
∀f
Iµ2
f dµ ≤
Iµ2 (f )2 + |∇f |2 dµ
first for µ the Bernoulli measure on two points, then by tensorization on
the hypercube (and µ a product of Bernoulli) and finally by the central
limit theorem on Rn for the standard Gaussian measure µ = µn2 . One of
the nice feature of the latter is that it tensorizes, i.e. the inequality in
dimension one implies the same inequality in any dimension. However it
appeared to be very hard to generalize tobobkov-houdre-96
other measures, at the notable
exception of the two-sided exponential µ1 [26]. The former Gaussian
result
ehrhard
[33]
and
is also established using a symmetrization argument
by
Ehrhard
barthe-maurey
a martingale approach by Barthe and Maurey [14]. An
other
proof
was
bakry-ledoux
given, using analytic techniques, by Bakry and Ledoux [5]. Those authors
also established an isoperimetric inequality of Gaussian type for general
measures under a curvature condition. Moreover, their technique allowed us
to prove the following result.
th:bcr
bcr-06,bcr-07
Theorem 4 (Barthe-Cattiaux-Roberto [12, 13]). There exists K > 0 such
that for any α ∈ [1, 2],
KIµα ≤ Iµ∞
≤ Iµα .
α
bobkov-houdre-96
This result was proved by Bobkov and
√ Houdré [26] for the two-sided
can
exponential measure µ1 , with K = 1/(2 6). A more general version √
bcr-07
−1 −Φ(x)
dx with Φ convex, th:bcr
be found in [13, Theorem 15] for dµ(x) = ZΦ e
Φ
concave and few mild technical assumptions. Sodinsodin
exploited Theorem 4 in
his study of the isoperimetric problem on ℓp balls [56]. We shall sketch the
proof of this theorem in the next section. But before that, let us complete
the picture:
what
happens for α > 2 and α < 1?
barthe-maurey
barthe-02,ros
In [14] (see [8, 55] for extensions), it is proved that if Iµ ≥ cIµ2 for some
c > 0, then
eq:product
(2)
Iµn ≥ cIµn2 = cIµ2 .
This
has applications in the isoperimetric problem for product of spheres
barthe-01bis
[6]. Using an argument based on the central limit theorem, it is possible to
prove that, starting
from I ≥ cIµ2 , in the limit Iµ∞ ≤ cµ Iµ2 (i.e. the other
eq:product µ
and its limit
direction than (2)). Hence, the sequence (Iµk )k is decreasing
barthe-02
compares to the Gaussian isoperimetric profile. See [8] for a control on the
speed of convergence in terms of what is called the isoperimetric dimension.
6
C. ROBERTO
eq:product
with a nice refinement
of the anaUsing an extension of (2) together bakry-ledoux
milman-07
lytic technique of Bakry and Ledoux [5] (first used in [50]) and capacity
arguments, E. Milman proved the following result.
milman-O8
th:milman
Theorem 5 (E. Milman [51]). Let I : [0, 1] → R+ be a continuous concave
function satisfying I(0) = 0 and I(1 − a) = I(a) for any a ∈ [0, 1]. Assume
that there exists c > 0 such that for any 0 < s ≤ t ≤ 12
eq:milman
(3)
I(s)
I(t)
≤c
.
Iµ2 (s)
Iµ2 (t)
Let µ be a probability measure on R such that Iµ ≥ I on [0, 1]. Then, there
exists a constant c′ such that for any n,
Iµn ≥ c′ I
on [0, 1].
This recovers
the previous theorem. Indeed it is easy to check that µα
eq:milman
= Iµα . Note that it is probably feasible tobcr-07
approach the
satisfies (3) with Ith:milman
proof of Theorem 5 by pushing further the techniques used in [13] but at theth:milman
price of some more assumptions and technicalities.
In this sense Theorem 5
th:bcr
is somehow more intrinsic than Theorem 4.
Observe that E. Milman’s theorem deals with measureseq:milman
“between” exponential and Gaussian, in the sense that Assumption (3) imposes that
the measure µ is below the Gaussian and the concavity assumption on the
isoperimetric profile forces µ to be above the exponential.
In fact, dimension free results, as in the two previous theorems, can only
be obtained for measures “between” exponential and Gaussian. That the
distribution has tails heavier than Gaussian was explained by the central
talagrand-91
limit argument above. On the other hand, as observed by Talagrand [58]: if
µ is a probability measure on R such that there exists h > 0 and ε > 0 for
which for any n ≥ 1, and all A ⊂ Rn with µn (A) ≥ 1/2, one has
1
+ε
2
then µ has at least exponential tails, that is there exists C1 , C2 > 0 such
that µ([x, ∞)) ≤ C1 e−C2 x , x ∈ R. This implies that for measures with
heavy tails, the isoperimetric profile of µn has to depend on the dimension
and goes to 0 as n goes to infinity. Using transportation of mass techniques,
we have the following result in this direction.
µn (A + [−h, h]n ) ≥
th:cggr
cggr
Theorem 6 (Cattiaux-Gozlan-Guillin-Roberto [31]). Consider a symmetric probability measure dµ(x) = ZΦ−1 e−Φ(x) dx, on R, with Φ concave on
[0, ∞). Then, for any n and any Borel set A ⊂ Rn ,
min (a, 1 − a)
′
−1
.
Iµn (a) ≥ 2nFµ ◦ Fµ
24n
This theorem gives a lower bound
on the isoperimetric profile of Iµn .
bcr-05
An upper bound can be found in [11] using the following strategy: since
ISOPERIMETRY FOR PRODUCT OF PROBABILITY MEASURES
7
(An )h ⊂ (Ah )n (where An is the Cartesian product of A), we have
µns (∂A) ≤ nµ(A)n−1 µs (∂A).
Hence, by the very definition of the boundary measure, for all a ∈ (0, 1),
Iµn (an ) ≤ nan−1 Iµ (a).
Now we illustrate this on two explicit examples. For the sub-exponential
th:cggr
−|x|α /α
dx, α ≤ 1, we get from Theorem 6 (for the
type law dµα (x) = 1e
2α α Γ(1+(1/α))
lower bound) and the above strategy (for the upper bound) that there exists
two constants c1 , c2 > 0 such that for any n
1
1− 1
α
1
n 1− α
n
n
≤ Iµα (a) ≤ c2 p log
, p = min(a, 1−a).
c1 p log
log
p
p
log(1/p)
α
For Cauchy-type distributions of the form dmα (x) = 2(1+|x|)
1+α dx, α > 0,
we get that there exists two constants c1 , c2 > 0 such that for any n
1− 1
α
1
1
1+ α
p
log
p
p
,
p = min(a, 1 − a).
c1 1 ≤ Imnα (a) ≤ c2
1
nα
nα
Note that in both cases the dependence in n is of the correct order.
As a summary
we end this section with a “graph” suggested to us by
barthe-private
Franck Barthe [10]. The first coordinate is the dimension n, the second
corresponds to α > 0. We represent the behavior of Iµnα as a function of n.
For any α > 0, Iµnα is non-increasing (as a function of n). It decays to the
Gaussian isoperimetric profile (up to constants) when α > 2, it is constant
for α = 2, almost constant for α ∈ (1, 2) and goes to 0 when α < 1.
α
cIµ2 ≤ lim Iµnα ≤ c′ Iµ2
n→∞
2
1
0
1 2 3
 Iµ2 = Iµ22 = · · · = Iµ∞
2

KIµα ≤ Iµ∞
≤ Iµα
α



I n ↓0
 µα
n
th:bcr
4. Analytic proof of Theorem 4
th:bcr
two
In this section we shetch the proof of Theorem 4. It mainly relies on
bakry-emery,bakry
ingredients, one based on the celebrated Γ2 calculus of Bakry-Emery [4, 3],
and the other one on functional inequalities.
P
The diffusion operator Lf (x) = ∆f (x)− ni=1 Φ′ (xi )∂i f , acting on smooth
functions on Rn , is symmetric in L2 (µn ), where dµ(x) = ZΦ−1 e−Φ(x) dx. Let
8
C. ROBERTO
bakry-ledoux
(Pt )t≥0 be its associated Markov semi-group. The following Lemma [5]
shows how the semi-group encodes some of the geometrical aspects of the
measure. Its proof heavily relies on the commutation property of the semigroup and the gradient of a function, which in turn comes from Γ2 calculus.
lem:bl
eq:bl
eq:p
bakry-ledoux
Lemma 7 (Bakry-Ledoux [5] Lemma 4.2). Let dµ(x) = ZΦ−1 e−Φ(x) dx be a
probability measure on R. Assume that Φ is convex and C 2 . Then, for any
n, any Borel set A ⊂ Rn with µn (A) ≤ 1/2,
√ Z
2
2
n
n
n
(4)
µs (∂A) ≥ √ µ (A) − (Pt 1A ) dµ .
t
n
A similar result holds
≥ 1/2. The next step in the proof
R when 2µα (A)
consists in controlling (Pt 1A ) dµn . This will be achieved using functional
inequalities. Let us explain the strategy on the simple example of the twosided exponential
measure µ1 and the Poincaré inequality, as observed by
ledoux-94
Ledoux [43]. It is well known that the two-sided exponential satisfies the
following (optimal) Poincaré inequality
2
Z
Z Z
n
(5)
f − f dµ1 dµn1 ≤ 4 |∇f |2 dµn1 .
P
Here |∇g|2 := ni=1 (∂i g)2 is the square of the Euclidean norm of the graR
R
dient. Consider u(t) = (Pt f )2 dµn1 with f satisfying f dµn1 = 0. Then,
using the integration by parts formula and the Poincaré inequality, we have
Z
Z
1
′
n
u (t) = 2 Pt f LPt f dµ1 = −2 |∇Pt f |2 dµn1 ≤ − u(t).
2
Hence, after integration, and by approximation of 1A , we get
2
2
Z Z
Z Z
Pt 1A − 1A dµn1 dµn1 ≤ e−t/2
1A − 1A dµn1 dµn1 .
After few rearrangements we arrive at
Z
(Pt 1A )2 ≤ µn1 (A)2 + e−t/2 µn1 (A) (1 − µn1 (A)) .
Hiding some technicalities (the potential Φ(x) = |x| is not twice differentiable and should be replaced by some smooth version) and applying Bakry
and Ledoux’s Lemma we finally get that for any A with µn1 (A) ≤ 1/2,
′
c 1 − e−c t
√
µn1 (A) (1 − µn1 (A)) .
(µ1 )ns (∂A) ≥
t
for some constant c, c′ > 0 (coming th:bcr
from the technical approximations).
4
for µ1 . The latter is known as the
Optimizing over t leads
to
Theorem
cheeger
Cheeger inequality [32].
eq:bl
One of the main feature
of this approach is that Inequality (4) and the
eq:p
Poincaré inequality (5) are dimension free.
ISOPERIMETRY FOR PRODUCT OF PROBABILITY MEASURES
9
Note that isoperimetric inequalities imply (without any assumptions)
L2
cheeger,mazja
functional inequalities of Poincaré type, by the co-area formula [32, 49].
The converse is false in general and one has to add some (e.g. convexity)
assumptions.
th:bcr
In order to prove Theorem 4 we have to find an appropriate dimension
free functional inequality
to the family of measures µα , and then
R associated
2
n
to derive a bound on (Pt 1A ) dµα . This will be achieved using the socalled super Poincaré inequalities. Unlike the standard Poincaré inequality,
those inequalities do not tensorize in general. So one has to prove first an
intermediate family of inequalities, called Latala and Oleszkiewicz inequalities, that do tensorize, and then derive, via capacity arguments, the super
Poincaré inequality along the scheme:
Latala and Oleszkiewicz inequality in dimension 1
⇒ Latala and Oleszkiewicz inequality in dimension n
⇒ Capacity-measure inequality
⇒ Super-Poincaré inequality in dimension n.
Now we introduce those inequalities. A measure µn is said to satisfy a Latala
and Oleszkiewicz inequality (respectively a super Poincaré inequality) if for
some C > 0, every smooth f : Rn → R satisfies
eq:lo
(6)
sup
p∈(1,2)
R
R
2/p
Z
f 2 dµn −
|f |p dµn
≤ |∇f |2 dµn
T (2 − p)
with T (x) = Cx2(1− α ) , respectively
1
eq:sp
(7)
Z
f 2 dµn ≤ β(s)
Z
|∇f |2 dµn + s
Z
|f |dµn
2
∀s ≥ 1
1
2(1− α
).
with β(s) = C/ log(1 + s)eq:lo
beckner
The case T (x) = x in (6) was proved by Beckner [18] for the measure µ2 .
This
corresponds
to the stronger logarithmic Sobolev inequality of Gross
gross-75
gross-93,ane
eq:lo
[37] (see also [38, 2]). Latala and Oleszkiewicz proved that Inequality (6)
1
holds with T (x) = Cx2(1− α ) for µnα and deduced some concentration
results.
barthe-roberto
This was then re-proved using Hardy type inequalities in [15].
latala-oleszkiewicz
The
two first steps of the previous scheme are thus a consequence of [42]
barthe-roberto
(or [15]).
mazja
Using the notion of capacity introduced
by Maz’ja [49] and revisited in
barthe-roberto
the setting of probability measuresbcr-06
in [15], the two last steps of the previous
bcr-07
[12,
Theorem
18
and
Lemma
19]
and
[13,
scheme follow (see more precisely
wang-05,wang-book
Corollary 6], see also
Wang [61, 60]). Hence the measure µnα , α ∈ [1, 2],
eq:sp
satisfies Inequality (7) with a constant C independent of n.
10
C. ROBERTO
R
2
n
Now let u(t) = (P
t f ) dµα . As for the Poincaré inequality, after differeq:sp
entiation, thanks to (7) we have for any s ≥ 1,
Z
2 !
Z
2
′
2
n
n
u (t) = −2 |∇Pt f | dµα ≤ −
u(t) − s
|f |dµα
.
β(s)
Hence, by approximation of 1A (with A ⊂ Rn a Borel set), we get
Z
(Pt 1A )2 dµnα ≤ e−2t/β(s) µnα (A) + s 1 − e−2t/β(s) µnα (A)2 .
lem:bl
Hiding again some technicalities, we can use Lemma 7 to obtain that
√
2
n
∀t > 0, ∀s ≥ 1.
(µα )(∂A) ≥ √ µnα (A) (1 − sµnα(A)) 1 − e−2t/β(s)
t
Choosing s = 1/(2µnα (A)) and t = β(s)/2 leads to
1− 1
α
1
n
n
(µα )(∂A) ≥ cµα (A) log n
≥ c′ Iµα (µnα (A))
µα (A)
for every A such that µnα (A) ≤ 1/2 and some constant c′ > 0 that does not
depend on n. Since a similar result
holds true for A with µnα (A) ≥ 1/2, this
th:bcr
achieves the proof of Theorem 4.
5. Concentration of measure phenomenon
In this section we derive a somehow new concentration result ledoux-concentration
from the
isoperimetric inequality for the Cauchy distributions. We refer to [46] for a
detailed account on the topic of concentration of measure phenomenon and
its applications. The connection between isoperimetry and concentration is
given in the following result.
prop:conc
Proposition 8. Let dµ(x) = ZΦ−1 e−Φ(x) dx be a probability measure on R.
Let v : R → [0, 1] be an increasing differentiable function. The following are
equivalent
(i)
Iµn ≥ v ′ ◦ v −1 .
(ii) For every h > 0 and every Borel set A ⊂ Rn ,
µn (Ah ) ≥ v v −1 (µn (A)) + h .
ledoux-concentration
eq:conc
See [46, Proposition 2.1] for the proof. For the Gaussian measure µn2 ,
2
since Fµ2 (0) = 21 and Fµ2 (h) ≥ 1 − e−h /2 , Sudakov-Tsirel’son and Borell’s
Theorem together with the latter proposition ensure that, for any n and any
Borel set A ⊂ Rn with µn2 (A) ≥ 1/2,
1
−h2 /2 .
(8)
µn2 (Ah ) ≥ Fµ2 Fµ−1
2
2 + h ≥ Fµ2 (h) ≥ 1 − e
n , α ≥ 2 can be obtained using infSuch a bound for the
measures µbobkov-gotze,bobkov-gentil-ledoux
α
maurey-91
convolution method [48] (see also [25, 24]). Also, the celebrated Herbst
ISOPERIMETRY FOR PRODUCT OF PROBABILITY MEASURES
11
argument implies the former Gaussian
concentration result via the logaane,ledoux-99
introduced a
rithmic Sobolev inequality, see [2, 45]. Bobkov and Ledoux
bobkov-ledoux
modified version of the logarithmic Sobolev inequality [20]
in order to imtalagrand-91
Those
prove an exponential concentration result of Talagrand [58] for µn1 . gentil-guillin-miclo-05,gentilmodified inequalities were generalized by Gentil, barthe-roberto-08
Guillin and Miclo [34, 35]
and re-proved, using
Hardy type marton
inequalities, in [16] and by transportation
gozlan
of mass technique [36] (see also [47]), leading to new concentration results
for µα , α ≥ 1.
eq:conc
A concentration result of the type (8) is, a priori, weaker than the isoperimetric inequality.
However, eq:conc
under a convexity assumption, very recently E.
milman-09
Milman [52] proved that (8) (with a general function replacing h2 /2) is
equivalent to an isoperimetric inequality.
bcr-05
Here we would like to focus on measures with heavy tails. In [11], some
concentration results of Lipschitz functions for product of measures with
heavy tails are aida
obtained, using an induction technique of Aida, Masuada
th:cggr
and Shigekawa [1]. Here we would like to take adventage of Theorem 6.
Proposition 9. Let dµ(x) = ZΦ−1 e−Φ(x) dx be a symmetric probability measure on R with Φ concave on [0, ∞). Then, for any n, any Borel set A ⊂ Rn
and any h > 0 with µn (A) ≥ 1/2,
1
h
µn (Ah ) ≥ 1 − 24nFµ Fµ−1
−
.
48n
12
n
Proof. Let Ψ(h) = Fµ−1 1−µ24n(Ah ) with A ⊂ Rn a Borel set satisfying
th:cggr
µn (A) ≥ 1/2. Thanks to Theorem 6 we have
Ψ′ (h) = −
Hence,
Ψ(h) = Ψ(0) +
Z
h
0
µns (∂Ah )
1
≤− .
n (A )
1−µ
−1
h
12
24nFµ′ ◦ Fµ
24n
h
Ψ (u)du ≤ Ψ(0) −
= Fµ−1
12
′
The expected result follow by monotony of Fµ .
1 − µn (A)
24n
−
h
.
12
We illustrate this proposition on the simple example of the Cauchy disα
tributions dmα (x) = 2(1+|x|)
1+α dx, α > 0. For r ≤ 0, we have Fmα (r) =
1
2(1−r)α . Hence, after simple computations (left to the reader), using the fact
that µn (A) ≥ 1/2, we get that for h = tn1/α (which is the correct scaling),
mnα (Atn1/α ) ≥ 1 −
bcr-05
121+α
.
tα
This
improves [11, Inequality (5.12)] of a logarithmic factor. As explained
bcr-05
in [11, Inequality (5.16)] on the explicit example A = (−∞, r]n with r =
12
C. ROBERTO
−1 (a1/n ) determined so as to have mn (A) = a ≥ 1/2, the latter estimate
Fm
α
α
is of correct order, since for this particular choice of A,
mnα (Atn1/α ) ≤ 1 −
Cα
tα
for some constant Cα > 0.
With more efforts, similar computations can be done for
concave pobcr-05
tentials of power type, recovering the concentration result [11, Proposition
6.4]: under mild assumptions on Φ : [0, ∞) → R+ concave, the measure
dµ(x) = ZΦ−1 e−Φ(|x|) dx satisfies
′
µn (Ah ) ≥ 1 − Ce−cΦ ◦Φ
−1 (max(Φ(h),log n))
∀h > 0
for any Borel set A ⊂ Rn with µn (A) ≥ 1/2 and some constants c, C > 0.
6. Final remarks and comments
We end this note with few comments on some recent works related to the
isoperimetric problem. Also we give some open directions..
Remark first that the isoperimetric problem changesbobkov-99
if one changes the
[22]
for a use of the
distance in the
definition
of
the
enlargement
A
.
See
h
bobkov-zegarlinski,milman-sodin,milman-O8
ℓ∞ distance, [29, 53, 51] for a use of the ℓp distance.
To deal with isoperimetric inequalities for non-product measures in not
an easy task. The analytic technique presented in this note
is one way in
milman-sodin
this direction. We mention the paper by Milmanhuet
and Sodin [53] around unifor spherically symmetformly log-concave probability measures, Huet [39]
bobkov-07
ric log-concavebarthe-wolff
probability measures and Bobkov [23] for convex probability
measures. In [17], an isoperimetric inequality is found for a canonical Gibbs
measure.
In order to go futher in the study of isoperimetric inequalities, it would
be certainly very interesting, and also very
challenging, to obtain some funcbobkov-houdre-96
tional inequalities. Indeed, it is known [26] that on the one hand the measure
µ1 satisfies
Z p
Z p
2
1 + f dµ1 ≤
1 + C(f ′ )2 dµ1
for all smooth
f : R → R, and on the other hand that the Gaussian measure
bobkov-97,bakry-ledoux
µ2 satisfies [21, 5]
Z
Z q
Iµ2
Iµ2 (f )2 + (f ′ )2 dµ2
∀f.
f dµ2 ≤
Hence it would be natural to have an intermediate (interpolating) inequality
for the measure µα , α ∈ [1, 2], involving Iµα . In particular, the two previous inequalities tensorize, and so one could get infinite dimension results
studying only
a one dimensional inequality. This direction would re-prove
th:bcr
Theorem 4. Also it would probably give some hints on how to deal with
non-product measures such as Gibbs measures.
ISOPERIMETRY FOR PRODUCT OF PROBABILITY MEASURES
13
Also, a generalization of the semi-group technique presented in this note
to other distance in the definition of the enlargement Ah would lead to very
interesting problems in non-linear analysis.
Finally kls
let us mention the very deep conjecture by Kannan, Lovász and Simonovits [40]. This conjecture asks for the existence of a universal constant
c > 0 such that for any dimension n, any n-dimensionalRlog-concave probability measure ν in the isotropicR position (i.e. such that < x, θ >2 dν(x) = 1
for any unit vector θ, and xdν(x) = 0) satisfies the following Cheeger
inequality
Iν (a) ≥ c min(a, 1 − a).
milman-07
Equivalently, thanks to a result by E. Milman [50], the conjecture asks for the
existence of a universal constant C > 0 such that for any dimension n, any
n-dimensional log-concave probability measure ν in the isotropic position
satisfies the following (a priori weaker) Poincaré inequality
Z
2
Z
Z
2
f dν −
f dν ≤ C |∇f |2 dν.
barthe-05
This conjecture would lead to many applications in convex geometry [9].
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ane
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bcr-06
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Cyril ROBERTO, Laboratoire d’Analyse et Mathématiques AppliquéesUMR 8050, Universités de Paris-est Marne la Vallée, 5 Boulevard Descartes,
Cité Descartes, Champs sur Marne, 77454 Marne la Vallée Cedex 2, FRANCE.
E-mail address: [email protected]