Large deviations and isoperimetry over convex probability measures with heavy tails

Large deviations and isoperimetric inequalities are considered for probability distributions, satisfying convexity conditions of the Brunn-Minkowski-type.


Introduction
A Radon probability measure µ on a locally convex space L is called κ-concave, where −∞ ≤ κ ≤ 1, if for all Borel subsets A, B of L with positive measure and all t ∈ (0, 1), where tA + (1 − t)B = {ta + (1 − t)b : a ∈ A, b ∈ B} denotes the Minkowski average, and where µ * stands for the inner measure. The mean power function appearing on the right hand side of (1.1), is understood as u t v 1−t when κ = 0, and as min{u, v} when κ = −∞. Inequality (1.1) is getting stronger, as κ increases, so the case κ = −∞ is the weakest one, describing the largest class in the hierarchy of the so-called convex probability measures (according to C. Borell's terminology, or hyperbolic measures, according to V. D. Milman).
In view of the homogeneity of M , the definition (1.1) also makes sense without the normalizing assumption µ(L) = 1. For example, the Lebesgue measure on L = R n or its restriction to an arbitrary convex set of positive Lebesgue measure is 1 n -concave (by the Brunn-Minkowski inequality). The case κ = 0 corresponds to the family of log-concave measures. The n-dimensional Cauchy distribution is κ-concave for κ = −1. Actually, many interesting multidimensional (or infinite dimensional) distributions are κ-concave with κ < 0, cf. (1) for more examples.
The dimension-free parameter κ may be viewed as the one characterizing the strength of convexity. A full description and comprehensive study of basic properties of κ-concave probability distributions was performed by C. Borell (1; 2); cf. also H. J. Brascamp and E. H. Lieb (12). As it turns out, such measures inherit a number of interesting properties from Gaussian measures, such as 0-1 law, integrability of (functions of) norms, absolute continuity of distributions of norms, etc. In this note we consider some geometric properties of κ-concave probability measures, which allow one to study large deviations of functionals from a wide class including arbitrary norms and polynomials. As one of the purposes, we try to unify a number of results, essentially known in the log-concave case, and to explore the role of κ in various inequalities.
For simplicity, we always assume L is finite-dimensional, although many dimension-free properties of such measures can easily be extended to infinite dimensional spaces (perhaps, under further mild assumptions about the space L). With every Borel measurable function f on L we associate its "modulus of regularity" δ f (ε) = sup x,y∈L mes{t ∈ (0, 1) : | f (tx + (1 − t)y) | ≤ ε |f (x)|}, 0 ≤ ε ≤ 1, (1.2) where mes stands for the Lebesgue measure. The behaviour of δ f near zero is connected with probabilities of large and small deviations of f . Moreover, the corresponding inequalities can be made independent of µ. This may be seen from the following: Theorem 1.1. Let f be a Borel measurable function on L, and let m be a median for |f | with respect to a κ-concave probability measure µ on L, κ < 0. For all h ≥ 1, where the constant C k depends on k, only.
This inequality may further be refined to reflect a correct behaviour as κ → 0, cf. Theorem 5.2 below. In the limit log-concave case κ = 0, the refined form yields an exponential bound with some universal c > 0.
For example, δ f (ε) ≤ 2ε for any norm f (x) = x , and then (1.3)-(1.4) correspond to a wellknown result of C. Borell (1). In the case of polynomials of degree at most d, one has δ f (ε) ≤ 2d ε 1/d , and then (1.4) represents a slightly improved version of a theorem of J. Bourgain (11), who studied Khinchin-type inequalities over high-dimensional convex bodies. In the general log-concave case, the bound (1.4) was recently obtained by different methods by F. Nazarov, M. Sodin and A. Volberg (25) and by the author in (9). While the approach of (25) is based on the bisection technique, here we follow (9) to extend the transportation argument, going back to the works of H. Knothe (17) and J. Bourgain (11).
In the second part of this note, we consider isoperimetric and analytic inequalities and derive, in particular: Theorem 1.2. Let µ be a non-degenerate κ-concave probability measure µ on R n , −∞ < κ ≤ 1. Here µ + (A) stands for the µ-perimeter of A, defined by µ + (A) = lim inf ε↓0 1 ε µ{x ∈ R n \ A : |x − a| < ε, for some a ∈ A}.
Up to the factor c m , the right hand side of (1.5) describes the asymptotically worst possible behaviour with respect to µ(A).
In the log-concave case, the median m is equivalent to the mean |x| dµ(x), and (1.5) turns into the Cheeger-type isoperimetric inequality, first obtained by R. Kannan, L. Lovász and M. Simonovits for the uniform distribution µ in an arbitrary convex body K (up to a universal multiplicative factor, cf. (16)). In that case, m = m(K), the so-called volume radius of K, may considerably be smaller than the diameter of the body. Actually, when κ = 1 n , the inequality (1.5) is stronger than the Cheeger-type and resembles the usual isoperimetric inequality for the Lebesgue measure. The general log-concave case in (1.5) was treated in (8) with a different functional argument, which was afterwards modified and pushed forward by F. Barthe (4). Theorem 1.2 may equivalently be formulated in terms of large deviations of Lipschitz functions under the measure µ. It may also be related to various analytic inequalities. For example, in case κ < 0 it follows from (1.5) that for all locally Lipschitz functions f on R n with µ-median zero. Of independent interest are also weak Poincaré inequalities, which perfectly reflect important properties of probability measures with heavy tails.
The paper is organized as follows. In section 2 we recall Borell's characterization of the κconcavity and consider separately the one-dimensional case. Some useful results about triangular maps are collected in section 3. They are used in section 4 to derive a geometric inequality of dilation-type for κ-concave probability measures. Theorem 1.1 is proved in section 5, together with some refinements. Section 6 deals with Khinchin-type inequalities and the problem on small deviations. In section 7, we derive Theorem 1.2. Related analytic inequalities are discussed in section 8. Finally, in section 9 general convex measures with a compact support are shown to share a Poincaré-type inequality of L. E. Payne and H. F. Weinberger.

Characterizations of convex measures
As was shown by C. Borell (1; 2), any convex probability measure on R n has an affine supporting subspace L, where it is absolutely continuous with respect to Lebesgue measure on L. For any κ-concave measure µ, it is necessary that κ ≤ 1 dim(L) , unless µ is a delta-measure. More precisely, when L = R n (this may be assumed in further considerations), the following characterization holds.
n . An absolutely continuous probability measure µ on R n is κconcave if and only if it is concentrated on an open convex set K in R n and has there a positive density p, which is κ(n)-concave in the sense that, for all t ∈ (0, 1) and x, y ∈ K, In particular, p must be continuous on the supporting set K.
The family of all full-dimensional convex probability measures µ on R n is described by (2.1) with κ(n) = − 1 n . The case κ = 1 n is only possible when p is constant, i.e., when K is bounded and µ is the uniform distribution in K.
If κ > 0, µ has to be compactly supported. For κ = 0, the density function must decay exponentially fast at infinity, that is, for some positive C and c, we have p(x) ≤ Ce −c|x| , for all x ∈ K. If κ < 0, the density admits the bound To see this, define the function p to be zero outside K. By Lemma 2.1, the sets of the form K(λ) = {x ∈ K : p(x) > λ} are convex. Since 1 ≥ K(λ) p ≥ λ mes(K(λ)), they are bounded for all λ > 0. Hence, p(x) → 0, as |x| → +∞.
Secondly, p is bounded. For, in the other case, pick a sequence x ℓ ∈ K, such that p(x ℓ ) ↑ +∞, as ℓ → ∞. Since K(λ) are bounded, we may assume x ℓ has a limit point x 0 ∈ clos(K). Let x 0 = 0 (without loss of generality). By (2.1) with x = x ℓ and s = 1 − t, we have p(tx ℓ + sy) ≥ M −1/n (p(x ℓ ), p(y)) ↑ s −n p(y), so p(sy) ≥ s −n p(y), for any y ∈ K and s ∈ (0, 1). But p(sy) dy = s −n p(y) dy, which is only possible when p(sy) = s −n p(y). The latter would imply that p is not integrable.
Finally, put A = sup x p(x) with the assumption that the sup is asymptotically attained at x 0 = 0. By the first step, choose r > 0 large enough so that p(x) ≤ 1 2 A, whenever |x| ≥ r. By Lemma 2.1, the function g( This argument also shows that p is compactly supported in the case κ > 0. Then the function g is concave, so, for λ ≥ 1, it satisfies g(λx) , where |x| = r and λx ∈ K, as before. Since g ≥ 0, necessarily λ ≤ λ 0 = 1 1−2 −κ(n) , and this means that K is contained in the Euclidean ball of radius at most λ 0 r.
In dimension 1, one can complement Lemma 2.1 with another characterization of the κ-concavity, which may be useful for the study of isoperimetric and large deviations inequalities. Let a probability measure µ be concentrated on some finite or infinite interval of the real line, say (a, b), and have there a positive, continuous density p. With it, we associate the function I(t) = p(F −1 (t)), defined in 0 < t < 1, where F −1 : (0, 1) → (a, b) denotes the inverse of the distribution function F (x) = µ((−∞, x]), a < x < b. Up to shifts, the correspondence µ → I is one-to-one between the family of all such measures µ and the family of all positive, continuous functions I on (0, 1). If the median of µ is at the origin, then the measure may uniquely be determined via the associated function by virtue of the relation F −1 (t) = t 1/2 ds I(s) . Lemma 2.2. A non-degenerate probability measure µ on the real line is κ-concave, −∞ < κ < 1, if and only if its associated function I is such that I 1/(1−κ) is concave. The measure is convex, if and only if the function log(I) is concave on (0, 1).
For an example, let us start with κ ≤ 1 and define a symmetric probability measure µ κ on the real line by requiring that its associated function is I κ (t) = (min{t, 1 − t}) 1−κ . By Lemma 2.2, µ κ is κ-concave. Its distribution function is given by If κ = 1, we obtain a uniform distribution on the interval [− 1 2 , 1 2 ]. When −∞ < κ ≤ 0, µ k is not supported on a finite interval. If κ = 0, we obtain the two-sided exponential distribution with density p(x) = 1 2 e −|x| . If κ < 0, the tails 1 − F κ (x) behave at infinity like 1 x α , α = − 1 κ . Let us note that, for any two probability measures µ and ν on the real line, having positive, continuous densities on their supporting intervals, the corresponding associated functions I and J satisfy I ≥ cJ (c > 0), if and only if the (unique) increasing map T , which pushes forward ν into µ, has a Lipschitz semi-norm T Lip ≤ 1 c . This observation can be used when µ is κ-concave and ν = µ k . By Lemma 2.2, where p is density of µ and m is its median. Hence: A similar observation will be made in the n-dimensional case, cf. Corollary 8.1.

Triangular maps
Here we recall definitions and sketch a few facts about triangular maps that will be needed for the proof of Theorem 1.1. In the convex body case such maps were used by H. Knothe (17) to reach certain generalizations of the Brunn-Minkowski inequality.
A map T = (T 1 , . . . , T n ) : A → R n defined on an open non-empty set A in R n is called triangular, if its components are of the form It is called increasing, if every component T i is a (strictly) increasing function with respect to the  where Equality (3.1) is a slight generalization of the usual change of the variable formula. The point of the generalization is that T is not required to be C 1 -smooth. Nevertheless, one may still use J(x) as a "generalized" Jacobian.
As a consequence, assume we have two absolutely continuous probability measures P and Q concentrated on A and B = T (A) and having there densities p and q, respectively. If they are related by the equality holding almost everywhere on A, then T must push forward P to Q. Indeed, applying (3.1) to f (y) = g(y)q(y) with Q-integrable g and using (3.2), we would get that That is, Q = P T −1 or Q = T # (P ).
Conversely, starting from P and Q, one may ask whether Q may be obtained from P as the image of T as above. The existence of a triangular map T satisfying (3.2) and with properties as in Lemma 3.1 requires, however, certain properties of P and Q. We say that a probability measure P concentrated on an open set A in R n is regular, if it has a density p which is positive and continuous on A, and for each i ≤ n − 1, the integrals If A is convex, then T (A) = B, so T is bijective. In the general case, B \T (A) may be non-empty, but has Lebesgue measure zero.
The components T i = T i (x 1 , . . . , x i ), 1 ≤ i ≤ n, of the map T can be constructed recursively via the relation for the conditional probabilities where X = (X 1 , . . . , X n ) and Y = (Y 1 , . . . , Y n ) are random vectors in R n with the distributions P and Q, respectively (when i = 1 these probabilities are unconditional). Thus, the (unique, increasing, continuous) function x i → T i (x 1 , . . . , x i ) is defined as the one, transporting the conditional distribution of X i given fixed x 1 , . . . , x i−1 to the conditional distribution of Y i given . This is where convexity of B and the regularity assumption are needed to ensure continuity of T and to justify derivation of the properties a)−b) in Lemma 3.2, cf. (9) for more details. Differentiating the equality (3.3) with respect to x i leads to with the convention that p 0 = q 0 = 1. Multiplying these relations by each other, we arrive eventually at p(x) = q(T (x))J(x).
To give some examples of regular measures in the above sense, for every set A in R n , together with the projections A i consider its sections A x 1 ,..., Without regularity assumption on the supporting set, we might be lead to certain singularity problems, so that part of the conclusions in Lemma 3.2 may fail. For example, the uniform distribution P on A = (0, 2) × (0, 1) ∪ (0, 1) × (0, 2) ⊂ R 2 is not regular. In this case, the density p 1 of the first coordinate is discontinuous at x 1 = 1. But continuity is necessary for the property a) in Lemma 3.2.
In the next section we apply Lemmas 3.1-3.3 to κ-concave measures µ restricted to regular subsets of R n .

Dilation
Given a Borel subset F of a (Borel) convex set K in R n and a number δ ∈ [0, 1], define We use mes to denote the one-dimensional Lebesgue measure on the (non-degenarate, closed) interval ∆ ⊂ R n . In the definition the requirement that x ∈ ∆ may be replaced with "x is an endpoint of ∆". Hence, the complement of F δ in K may be represented as The latter function is Borel measurable on K × K, so K \ F δ may be viewed as the x-projection of a Borel set in R n × R n . Therefore, the set F δ is universally measurable, and we may freely speak about its measure. Note also, if F is closed, then F δ is a closed subset of F (0 < δ < 1).
To illustrate the δ-operation, let us take a centrally symmetric, open, convex set B ⊂ K and put for any δ ∈ (0, 1), which is the complement to the dilated set B. So, relations between µ(F ) and µ(F δ ) belong to the family of inequalities of dilation-type.
As a basic step, we prove: In the case κ ≤ 0, the right-hand side of (4.1) is vanishing when µ(F δ ) = 0, so the requirement µ(F δ ) > 0 may be ignored. When κ = 0, (4.1) reads as In an equivalent functional form, which we discuss in the next section, the above inequality was obtained in (9). Actually, (4.2) can be improved to This is a correct relation obtained by F. Nazarov, M. Sodin and A. Volberg (25) in the spirit of localization results due to R. Kannan, L. Lovász and M. Simonovits (23), (16), with technique, going back to the bisection method of L. E. Payne and H. F. Weinberger (26) (cf. also (14)). This approach may be used to get many other sharp geometric inequalities for log-concave probability distributions. Although (4.2) is somewhat weaker, we do not know whether the more general inequality (4.1) can be sharpened in a similar manner as The refinement may improve absolute constants in some applications, e.g., the constant C κ in Theorem 1.1, which does not seem crucial. In some others, the refinement would be desirable.
Let us also mention that in the log-concave case (4.2) implies (4.3) for convex F . Indeed, one may automatically sharpen (4.2) by applying it to the product measure µ N on R nN and the product set . Letting first N → ∞ and then c → 0 yields (4.3).
In the non-convex case, we only have the inclusion (F N ) δ ⊂ (F δ ) N , and we do not know whether the above argument still works. However, it is to be emphasized that necessarily δ < 1 in the refined form (4.3). For, when δ = 1, we have F δ = K, which is larger than F . Hence, the factor c may not be removed in (4.1)-(4.2) with our definition of the δ-operation.
In the proof of Theorem 4.1, we use the following elementary lemma, known as (a partial case of) the generalized Hölder inequality.
Proof of Theorem 4.1. We may assume µ is full-dimensional, that is, absolutely continuous with respect to Lebesgue measure on R n . In particular, κ ≤ 1 n . Moreover, the supporting convex set K may be assumed to be open, and that µ has there a positive, κ(n)-concave density p.
When δ = 1, necessarily c = 0, and (4.1) is immediate. When δ = 0, since F δ is non-empty, necessarily µ(F ) = 1, and (4.2) is obvious. Indeed, taking a point x ∈ F δ , we get Integrating this equality over y with respect to some (any) absolutely continuous probability measure on K, having a positive density, we obtain that ν(F ) = 1, for some absolutely continuous probability measure ν on K. This implies µ(K \ F ) = 0. Now, let 0 < δ < 1. Take an open neighborhood G of F , containing in K, and take a regular subset A of K, such that µ(A ∩ F δ ) > 0 (e.g., a finite union of open balls). Denote by µ A the normalized restriction of µ to the set A. By Lemmas 3.2-3.3 and by the boundedness of p (cf. (2.2)), there is a continuous triangular map T : A → K, which pushes forward P = µ A to the measure Q = µ. Moreover, the components which is also continuous, triangular, with components that are C 1 -smooth with respect to x i -coordinates. Moreover, for all x ∈ A, its "generalized" Jacobian J t (x) satisfies where we applied an elementary inequality Consider an open set On the other hand, by Lemma 3.1 applied to T t , ). Moreover, combining (4.5) with the inequality of Lemma 4.2, we get that where we made use of (4.4) and of the homogeneity of the function M κ . Plugging this bound into (4.6), we get µ(D(t)) ≥ M κ (1, 1 µ(A) ) µ(B(t)), so, Now, we need a lower bound on the last term in (4.7). By the definition of the δ-operation, for all x ∈ G δ and y ∈ K, mes{t ∈ (0, 1) : Integrating this inequality over the set A ∩ G δ with respect to the normalized restriction ν of µ and interchanging the integrals gives On the other hand, it is bounded by 1. This actually implies that But this contradicts to (4.8). We may therefore conclude that . Recalling (4.7), we arrive at the bound Note that the function t → M (t) κ (u, v) is non-increasing for u ≤ v, so (4.9) also holds with t = t 0 . Finally, letting G ↓ F and approximating F δ with regular A's, so that µ(A ∩ F δ ) > 0 and µ((A \ F δ ) ∪ (F δ \ A)) → 0, we obtain (4.1). Theorem 4.1 is proved.
If µ is supported on a convex set K in R n , one may also apply (5.1) to functions defined on K (rather than on the whole space). Then in the definition (1. 2) of δ f the supremum should be taken over all points x, y in K.
An attempt to choose an optimal c in (5.1) complicates this inequality and in essense does not give an improvement. For applications, at the expense of some loss in constants, one may use Theorem 5.1 with c = 3 4 , for example.
Theorem 5.2. Let f be a Borel measurable function on R n , and let m be a median for |f | with respect to a κ-concave probability measure µ on R n , κ < 0. Then, for all h ≥ 1, such that If κ → 0, in the limit we obtain an exponential bound holding true for all log-concave probability measures µ.
Note the right-hand side of (5.1) does not exceed ( α C ) α δ f ( 1 h ) α , and we arrive at Theorem 1.
Proof. With notations as in proof of Theorem 5.2, let v = u κ . Then by (5.1), for all λ > 0 and Since the integer i ≥ 1 is arbitrary, the latter easily yields (5.4) with a different C.
For polynomials f with respect to the uniform distribution µ over an arbitrary convex body K in R n , such results were first obtained by J. Bourgain (11). In this case κ = 1 n is positive, and one may hope to get an additional information about the distribution of f in terms of κ and δ f (where the dependence on the dimension might be hidden in κ). Indeed, by Theorem 5.1 with holding true whenever 0 ≤ ε ≤ 1 and 0 ≤ λ < ess sup |f |. By Chebyshev's inequality, the left-hand side is bounded by E µ |f |/(λε). Letting λ → ess sup |f |, we arrive at

In particular:
Corollary 5.4. Let µ be a κ-concave probability measure on R n with κ > 0. Then, for any µ-integrable function f on R n , where ε k ∈ (0, 1] is chosen so that δ f (ε k ) ≤ κ 2 , and where C is universal.
For example, for any norm f (x) = x , we have sup x∈K x ≤ C κ K x dµ(x), where K is the supporting convex set of µ (which has to be bounded in case κ > 0). At the expense of the constant on the right, the expectation may be replaced with the µ-median m of f . Precise relations have been studied by O. Guédon (15), who showed that, for any κ-concave probability measure µ on R n , 0 < κ ≤ 1, and any convex symmetric set B,

Small deviations. Khinchin-type inequalities
If f (x) = x is a non-degenerate finite semi-norm on R n , it is easy to check that δ f (ε) = 2ε 1+ε . Therefore, for any κ-concave probability measure µ on R n , κ < 0, by Theorem 1.1, where m is a median for f with respect to µ, and the constant C α depends on α, only. By a different argument, this inequality was already obtained by C. Borell in (1). A more precise bound, useful when κ is close to zero, is given in Theorem 5.2.
The estimate (6.1) implies that the moments f q = ( |f | q dµ) 1/q are finite for q < α and are equivalent to each other in the sense that with constants C depending on q, q 0 , and κ. For various applications, it is however crucial to know whether or not it is possible to make C independent of q 0 and thus to involve the case q 0 = 0 in (6.2). Note that in general represents the geometic mean of |f | (provided that f q < +∞, for some q > 0). Hence, in order to sharpen (6.2) by replacing f q 0 with f 0 one needs to derive also bounds on small deviations of f . For log-concave probability measures, this question was settled by R. Latala in (18) by showing that µ{ x ≤ mε} ≤ Cε, 0 ≤ ε ≤ 1, for some absolute constant C. Actually, this estimate implies (6.2) for a larger range q 0 > −1. A different argument was suggested in (7). Further refinements are due to O. Guédon (15), who also considered the behaviour of constants for κ-concave probablity measures with κ > 0. While the argument of (15) is based on the localization lemma of L. Lovász and M. Simonovits (23) one can still use the transportation argument to derive (6.2) for a large class of functionals f , including arbitrary norms and polynomials. Namely, from Theorem 5.1 we have the following simple corollary: Theorem 6.1. Let f be a Borel measurable function on R n , and let m be a median for |f | with respect to a κ-concave probability measure µ on R n , κ ≤ 0. Then, where the constant C κ depends on κ, only.
Corollary 6.2. Let f be a Borel measurable function on R n , such that δ f (ε) ≤ R ε r in 0 < ε < 1, for some R, r > 0. Then with respect to any κ-concave probability measure µ on R n , κ ≤ 0, where the constant C depends on R, r, q, and κ.
As an example, one may apply Corollary 6.2 to an arbitrary norm or polynomial f (x 1 , . . . , x n ) in n real variables of degree at most d ≥ 1. In the second case, f represents a polynomial on every line of degree at most d, so the maximal possible value of we have with respect to the Lebesgue measure on (0,1) that Since |t − z i | ≥ |t − |z i | |, the roots z i may be assumed to be real non-negative numbers. But, for any c ∈ (0, 1) and z > 0, the quantity mes{|t − z| ≤ cz} is maximized at z = 1 1+c and is equal to 2c 1+c . This gives δ f (ε) ≤ 2d ε 1/d , that is, the hypothesis of Corollary 6.2 with R = 2d and r = 1 d . Hence, with C depending on d, q and κ. For norms, the above estimate holds with d = 1.
7 Isoperimetry. Proof of Theorem 1.2 Now, we turn to geometric inequalities, relating µ-perimeter of sets to their µ-measures.
Lemma 7.1. (4) If µ is a non-degenerate κ-concave probability measure on R n , for any Borel set A in R n , any point x 0 ∈ R n and r > 0, where B r (x 0 ) is the Euclidean ball of radius r with center at x 0 .
In the log-concave case (κ = 0), inequality (7.1) should read as By virtue of Prékopa-Leindler's functional form of the Brunn-Minkowski inequality, (7.2) was derived in (7). The arbitrary κ-concave case was treated by F. Barthe (4), who applied an extension of Prékopa-Leindler's theorem in the form of Borell and Brascamp-Lieb. Inequality (7.1) was used in (4) to study the concept of the isoperimetric dimension for κ-concave measures with κ > 0.
Without loss of generality, one may state Lemma 7.1 with x 0 = 0. To make the proof of Theorem 1.2 to be self-contained, let us give a direct argument for (7.1), not appealing to any functional form. It is based on the following representation for the µ-perimeter, explicitely relating it to measure convexity properties. Namely, let a given probability measure µ on R n be absolutely continuous and have a continuous density p(x) on an open supporting convex set, say K. Then, for any sufficiently "nice" set A, e.g., a finite union of closed balls in K or the complement in R n to the finite union of such balls, where B r is the Euclidean ball of radius r with center at x 0 = 0. Indeed, the µ-perimeter is described as the (n − 1)-dimensional integral with respect to Lebesgue measure H n−1 on the boundary ∂A of the set A. Moreover, for H n−1almost all points x in ∂A, the outer normal unit vector n A (x) at x is well-defined, and for any r > 0, The complementĀ = R n \ A has the same boundary and the same µ-perimeter µ + (Ā), and we have nĀ(x) = −n A (x). Hence, forĀ the above relation reads as Summing the two relations, we arrive at (7.3).
In the case of a κ-concave µ, it remains to apply in (7.3) the original convexity property (1.1), and then we obtain (7.1).
Remark. More generally, one could start from a Brunn-Minkowski-type inequality of the form where F : (α, β) → (0, 1) is a C 1 -smooth monotone function with inverse F −1 and where we assume 0 < µ(A), µ(B) < 1. Then, a similar argument leads to with t = µ(A) and I(t) = F ′ (F −1 (t)). The case of a κ-concave µ corresponds to F (u) = u 1/κ , and then (7.4) reduces to (7.1). When F represents the distribution function of some symmetric probability measure on the line, (7.4) simplifies to For example, the famous Ehrhard's inequality states that the standard Gaussian measure µ on R n satisfies the Brunn-Minkowski-type inequality with the standard normal distribution function F = Φ. In this case, as easy to check, F −1 (µ(Br )) r → 1, as r → +∞, for any fixed n, and (7.5) becomes the Gaussian isoperimetric inequality That the Ehrhard inequality may directly be used to derive (7.6) in its "integral" form, µ(A + B r ) ≥ Φ(Φ −1 (µ(A)) + r), was noted by M. Ledoux (19).
Proof of Theorem 1.2.
In terms of the isoperimetric function I µ (t) = inf µ(A)=t µ + (A), associated to the given κ-concave probability measure µ, inequality (1.5) may be written as where c = c(κ) is a positive, continuous function, defined on the half-axis κ ≤ 1. Recall that m is defined to be a real number, such that µ{x : |x| ≤ m} = 1 2 . By symmetry, let 0 < t ≤ 1 2 . Introduce the tail function u(h) = µ{x : |x| > mh}, h ≥ 1. Choosing x 0 = 0 and r = mh in Lemma 7.1, one may write inequality (7.1) as Thus, our task is to properly estimate the right hand side of (7.7) by choosing somewhat optimal h. It will be conventient to consider separately the two cases.

Theorem 1.2 is proved.
8 Functional forms. Sobolev-type inequalities Theorem 1.2 admits the following equivalent formulation, involving the one-dimensional κconcave probability measures µ κ with the distribution functions F κ given in (2.3). As before, let µ be an arbitrary κ-concave probability measure on R n , −∞ < κ ≤ 1, and let m denote the µ-median for the Euclidean norm, viewed as a random variable on the probability space (R n , µ). Let us recall the argument, which is standard. The isoperimetric inequality (1.5) can be "integrated" over the parameter h > 0 to yield for any Borel set A in R n , such that µ(A) > 0, where F −1 κ is the inverse function. In turn, letting h → 0 in (8.1), we return to (1.5). For simplicity of notations, let κ ≤ 0, so that the supporting interval of µ k is the whole real line. Given a Lipschitz function f on R n , the sets A(a) = {x : f (x) ≤ a} satisfy A(a) + hB 1 ⊂ A(a + h). Hence, (8.1) implies where F is the distribution function of f under µ. Let f be non-constant modulo µ. Then, the interval {a : 0 < F (a) < 1} is non-empty, and, by (8.2), F strictly increases on it. Define the "inverse" function by putting Since it pushes forward the uniform distribution on (0,1) to the measure with the distribution function F , the map T = F −1 (F κ ) pushes forward µ k to the law of f . In terms of S = F −1 κ (F ), this map may also be defined as T (x) = min{a ∈ R : S(a) ≥ x}, x ∈ R.
It remains to check that T (x + ch) ≤ T (x) + h, for all x ∈ R and h > 0. Given S(a) ≥ x, we need to see that T (x + ch) ≤ a + h. Indeed, by (8.2), Thus, the map T has a Lipschitz semi-norm T Lip ≤ 1 c , and Corollary 8.1 is proved.
When κ < 0, it follows from (8.2) that any Lipschitz f with µ-median zero has a distribution with tails satisfying More delicate properties (in comparison with large deviations of Lipschitz functions) may be stated as Sobolev-type inequalities. First, we derive: For any smooth function f on R n with µ-median zero, where the constant C depends upon q and κ, only.
When κ ≥ 0, inequality (8.3) is also true with a suitable constant for the critical power q = 1 1−κ , and then it represents an equivalent analytic form for (1.5). However in case κ ≤ 0, when f 's approximate indicator functions of Borel sets, (8.3) turns into an isoperimetric inequality, which is close to, but still a little weaker than (1.5). Therefore, it would be interesting to look for other functional forms. As it turns out, there are suitable equivalent analytic inequalities in the class of weak Poincaré-type inequalities for L 1 -norm of f . In particular, one may use the following characterization: c) For any bounded, smooth function f on R n with µ-median zero, Here we use the usual notatation Osc µ (f ) = ess sup f − ess inf f for the total oscillation of f with respect to the measure µ.
The statement c) follows from b) by optimization over s > 0, and similarly the converse implication holds. As for the equivalence of a) and b), one may start more generally from the isoperimetric-type inequality with a positive, Borel function I = I(t) in 0 < t < 1, symmetric about t = 1 2 , and look for an optimal function β 1 in |f | dµ ≤ β 1 (s) |∇f | dµ + s Osc µ (f ) (8.5) in the same class of functions f as in b). Here, the smoothness requirement may be relaxed to being Lipschitz or locally Lipschitz. Note also, since |f | dµ ≤ 1 2 Osc µ (f ) for any f with median zero, only the values s ∈ (0, 1 2 ] are of interest in the inequality (8.5). Again, first assume f ≥ 0 has a finite Lipschitz seminorm and satisfies µ{f = 0} ≥ 1 2 . In view of the homogeneity of (8.5), also let ess sup f = 1. An application of the generalized coarea inequality leads to where the optimal function is In the general case, we may apply (8.5) to f + = max{f, 0} and f − = max{−f, 0}, which are non-negative, Lipschitz, and have median at zero. More accurately, we then have Adding the two inequalities and making use of Osc µ (f + ) + Osc µ (f − ) ≤ Osc µ (f + ), we arrive at (8.5) with β 1 (s), defined in (8.7).
Conversely, once we know that (8.5) holds, one may approximate indicator functions of Borel sets A by functions with finite Lipschitz semi-norm (e.g., as in (10), Lemma 3.5) to derive µ(A) ≤ β 1 (s)µ + (A) + s. The latter implies the isoperimetric inequality (8.4) with the function I, satisfying the same relation (8.6). Therefore, the optimal choice should be Now, let us return to an arbitrary κ-concave probability measure µ on R n and apply Lemma 8.3 with p = 1 − κ. Then we arrive at the following equivalent form for (1.5) with some (different) positive, continuous function c(κ).
Corollary 8.4. If κ < 0, for any bounded, smooth function f on R n with µ-median zero, and for all s > 0, It is easy to see that c(κ) may be chosen to be separated from zero, namely, to satisfy lim κ→0 c(κ) = c > 0, so (8.9) also includes the case κ = 0. Letting s → 0 in order to get rid of the oscillation term, we then arrive at the L 1 -Poincaré-type inequality for the class of log-concave measures µ. By Cheeger's argument, the latter is known to be equivalent to the isoperimetric inequality (1.5) with κ = 0, and is also known to imply the L 2 -Poincaré or the spectral gap inequality where Var µ (f ) = f 2 dµ − ( f dµ) 2 denotes the variance of f with respect to µ. The argument may easily be extended to involve more general inequalities such as (8.9). Let f ≥ 0 be a bounded, smooth function with µ-median zero. Applying (8.5) to f 2 and then Cauchy's inequality, we obtain that L 2 -norms of f and |∇f | satisfy This extends to the case of general f in the form of the weak Poincaré-type inequality with β 2 (s) = 4β 1 ( s 2 ) 2 . Hence, by Corollary 8.4, any κ-concave probability measure µ on R n with κ ≤ 0 satisfies for some positive continuous function c(κ) with an arbitrary bounded smooth f .

Remarks.
In connection with the slow convergence rates of Markov semigroups (P t ) t≥0 , associated to µ, weak Poincaré-type inequalities have been introduced by M. Röckner and F.-Y. Wang (27). They considered (8.10) as an additive form of inequalities of Nash-type, considered before by T. M. Liggett (21). See also the work by L. Bertini and B. Zegarlinski (6), where generalized Nash inequalities were studied for Gibbs measures.
Inequality (8.10) allows one to quantify the weak spectral gap property (WSGP, for short) in the sense of Kusuoka-Aida. As given in (3), the latter is defined as the property that f ℓ → 0 in µ-probability, as long as f ℓ 2 ≤ 1, f ℓ dµ = 0 and E(f ℓ , f ℓ ) → 0, as ℓ → ∞, where the Dirichlet form in our particular case is E(f, f ) = |∇f | 2 dµ. It is shown in (27) that WSGP is equivalent to (8.10) with β 2 (s) < +∞, for all s > 0. In terms of L 1 -convergence of P t another characterization was given by P. Mathieu (24).
Let us also mention that recently F. Barthe, P. Cattiaux, and C. Roberto have studied relationship between (8.10) and capacitary inequalities on metric probability spaces, and applied them to get concentration inequalities for product measures, involving dependence on the dimension, cf. (5).

Compactly supported measures. Localization
If a κ-concave probability measure µ on R n is compactly supported and its supporting set K is contained in some ball B r (x 0 ) of radius r, (7.1) yields, In particular, if K has dimension n and µ represents the normalized Lebesgue measure on K, Barthe's estimate (9.1) becomes 3) The right-hand side has a correct behaviour as µ(A) is getting small. In the other case, when µ(A) is of order 1 2 , this isoperimetric inequality may asymptotically be sharpened with respect to the dimension. Namely, by Theorem 1.2, for some universal c > 0, where m is the µ-median for the Euclidean norm (the volume radius of K).
For example, for the unit ℓ 1 -ball K = {x ∈ R n : |x 1 |+· · ·+|x n | ≤ 1}, we have r = 1, and if µ(A) = t is fixed, the right-hand side of (9.3) is convergent, as n → ∞, to 1 2 [ t log 1 t + (1 − t) log 1 1−t ]. Hence, there is no improvement over the log-concave case as in (9.2). On the other hand, the volume radius of K is of order 1 √ n , so the right-hand side of (9.4) is of order t √ n.
This difference may be explained by the fact that the minimal ball, containing the supporting set, is not optimal in (7.1). Nevertheless, (9.1) may still be used to recover a number of results such as a Cheeger-type isoperimetric inequality Note that, for any κ ∈ (−∞, 1], the function u(t) is critical in the inequality of the form u(t) ≥ c min{t, 1 − t}. Therefore, if µ is κ-concave, (9.1) implies (9.5) with constant In particular, for the range κ ≥ 0, we have a uniform bound on the isoperimetric constant, c κ (r) ≥ log 2 2r . In the body case, similar bounds in terms of the diameter d = diam(K), i.e., of the form I µ (t) ≥ c d min{t, 1 − t} were studied by many authors in connection with randomized volume algorithms, cf. e.g. (22), (13).
When µ is the normalized Lebesgue measure on K, this inequality with an optimal absolute factor in front of diam(K) 2 was obtained by L. E. Payne and H. F. Weinberger (26). It should be noted that in the class of log-concave probability measures on R n Cheeger-type isoperimetric and Poincaré-type inequalities are in essense equivalent. This has recently been shown by M. Ledoux (20), The proof of (9.7) requires to apply an additional localization technique in the form, proposed in (26) and developed in (23). First consider the one-dimensional case. Recall that a nondegenerated probability measure µ on the line with supporting interval K = (a, b) is called unimodal, if its distribution function F is convex on K, or concave on K, or if F is convex on (a, x 0 ) and concave on (x 0 , b), for some point x 0 ∈ K. If µ does not have an atom at x 0 , it is absolutely continuous and has a positive density p on K. Moreover, the associated function I(t) = p(F −1 (t)) is non-decreasing on (0, t 0 ) and non-increasing on (t 0 , 1), for some t 0 ∈ [0, 1]. We need the following: Lemma 9.2. Any unimodal probability distribution µ on the line, concentrated on a finite interval K = (a, b), satisfies the Cheeger-type isoperimetric inequality (9.7).
Proof. In dimension one, the best constant in µ + (A) ≥ c min{ µ(A), 1 − µ(A)} has a simple description as where p is the density of the absolutely continuous component of µ ( (10)). Hence, when the associated function I(t) is well defined, c = inf 0<t<1 min{t,1−t} . Now let µ be unimodal with a finite supporting interval (a, b). Without loss of generality, we may assume that µ has no atom and that it has a continuous density p, strictly increasing on (a, x 0 ] and strictly decreasing on [x 0 , b), for some a < x 0 < b. Hence, the function I is continuous on (0,1), is strictly increasing on (0, t 0 ) and strictly decreasing on (t 0 , 1), for some 0 < t 0 < 1. Let I −1 denote the inverse function for I, restricted to (t 0 , 1). By Chebyshev's inequality with respect to the Lebesgue measure λ on (t 0 , 1), for any ε ∈ (I(1−), I(t 0 )), 1 − I −1 (ε) = λ t ∈ (t 0 , 1) : 1 Letting ε = I(t) with t ∈ (t 0 , 1), we get that I(t) ≥ 1−t b−a . With a similar argument, I(t) ≥ t b−a , for any t ∈ (0, t 0 ). Hence, in both cases I(t) ≥ 1 b−a min{t, 1 − t} on (0,1), and the lemma follows. Note that, according to Lemmas 2.1 and 2.2, any convex probability distribution on the line is unimodal. Reduction to the one-dimensional case in Theorem 9.1 may be done with the help of the localization lemma of L. Lovász and M. Simonovits, which is stated below. The integrals in (9.8) are n-dimensional, while the integrals in (9.9) are taken with respect to the Lebesgue measure on ∆.
Proof of Theorem 9.1. Given a probability measure µ on R n , the inequality of the form µ + (A) ≥ c min{ µ(A), 1 − µ(A)} with fixed c > 0, such as (9.7), may equivalently be stated as µ(A h ) ≥ R h (µ(A)), h > 0, (9.10) in the class of all open sets A in R n . Here A h = {x ∈ R n : dist(A, x) ≤ h} denotes the closed h-neighborhood of A with respect to the Euclidean distance, and R h (t) = F (F −1 (t) + ch) with F being the distribution function of the probability measure on the line with density 1 2 e −|x| . It will be convenient to reformulate (9.10) as implication h > 0, 0 < t < 1. (9.11) Now, let µ be a convex, compactly supported probability measure. We may assume µ is absolutely continuous. Then, the supporting convex set K may be chosen to be open, and by Lemma 2.1, the density p(x) of µ may be chosen to be κ-concave on K with κ = − 1 n . Suppose that (9.7), that is, the property (9.11) with c = 1 diam(K) is not true for some A, t and h, so that µ(A) > t and µ(A h ) < R h (t). This may be written as Hence, the hypothesis (9.8) of Lemma 9.3 is fulfilled for u(x) = (1 A (x) − t)p(x) and v(x) = (R h (t) − 1 A h (x))p(x). Note these functions are lower-semicontinuous, since A is open and A h is closed. Therefore, one can find points a, b ∈ K and a non-negative affine function ℓ, defined on the interval ∆ = [a, b], such that (9.9) holds. The latter may be written as ν(A) > t and ν(A h ) < R h (t) (9.12) in terms of the probability measure ν on ∆ with density q(x) = Z p(x) ℓ(x) n−1 with respect to the Lebesgue measure on ∆, where Z is a normalizing constant. But, by Lemma 4.2, the function q is κ-concave on ∆ with κ = −1, so that by Lemma 2.1 for dimension one, ν is convex on ∆. Therefore, by Lemma 9.2, ν satisfies the Cheeger-type isoperimetric inequality (9.7) and thus ν(A h ) ≥ R h (ν(A)). On the last step the bound |∆| ≤ diam(K) was used. As a result, we obtain a contradiction to (9.12).
Theorem 9.1 is now proved.