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Dilation type inequalities for strongly-convex sets in weighted Riemannian manifolds

In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell's lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.

for any centrally symmetric convex subset K ⊂ R n and t ≥ 1, it follows from (1.1) that 2 t + 1 µ(R n \ (tK)) s + t − 1 t + 1 µ(K) s 1/s ≤ µ(R n \ K) (1.2) when µ(K) > 0 and µ(tK) < 1. This inequality is called the dilation inequality or mentioned as Borell's lemma, and applied to high-dimensional convex geometry (for instance, see [7], [18], [16], [10]). However, the inequality (1.2) is not optimal for a convex subset K with µ(K) ≤ 1/2. Indeed, for instance, when µ is log-concave, the inequality (1.2) is equivalent to the form 3) and the right hand side above goes to 0 as t → ∞ if and only if µ(K) > 1/2. Lovász and Simonovits gave an optimal dilation inequality for log-concave probability measures and centrally symmetric subsets [19,Theorem 2.8], and later Guédon [16] proved by the localization method that for any s-concave probability measure µ with 0 ≤ s ≤ 1/n, centrally symmetric convex subset K ⊂ R n and t ≥ 1 (with µ(tK) < 1 when s > 0). Moreover, the above dilation inequality was generalized for any Borel subset in R n by Nazarov, Sodin and Vol'berg [25], Bobkov [2], [3], Bobkov and Nazarov [6], and Fradelizi [13] as follows. Given a Borel subset A ⊂ R n and t ≥ 1, we define A t ⊂ R n as A t := A ∪ x ∈ R n there exists some interval I ⊂ R n such that x ∈ I and |I ∩ A| > 2 t + 1 |I| , (1.4) where | · | means the 1-dimensional Lebesgue measure. We may assume that x is an endpoint of I in the definition of (1.4). Note that A t is a Borel set and A 1 = A. In addition, when A is an open convex subset in R n , Fradelizi [13,Fact 1] showed that for any t ≥ 1, Therefore, when A is symmetric centered at a ∈ R n , then A t = t(A− a)+ a. In particular, when a = 0, A t coincides with tA, and hence we may consider the set defined by (1.4) as a generalization of the dilation for centrally symmetric convex subsets. For other detailed properties of the dilation defined as (1.4), see [13]. The following inequality is the dilation inequality on the dilation set defined by (1.4): given an s-concave probability measure µ on R n with s ≤ 1/n, it holds that for any Borel subset A ⊂ R n and t ≥ 1 (with µ(A t ) < 1 when s > 0). Note that the inequality (1.5) is sharp. Indeed, when µ s is the probability measure on R whose density with respect to the 1-dimensional Lebesgue measure is where (·) + := max{·, 0}, then µ s is s-concave and equality holds in (1.5) for any interval [0, b] ⊂ R with b > 0. We comment on the methods of the preceding studies. Bobkov [2], [3] showed a weak type of (1.5) using triangular maps in mass transport theory, and Bobkov and Nazarov [6] and Fradelizi [13] used the localization method to prove (1.5). Recently, this localization method was extended to weighted Riemannian manifolds by Klartag [17] through optimal transport theory (more precisely, this extension corresponds to the localization by Lovász and Simonovits [19] which is used in [6], however, Fradelizi used the "geometric" localization method (see [14], [15] for more information) in which we need to use the Krein-Milman theorem). Since the characterization of densities of s-concave probability measures on R n by Borell [7] implies that the s-concavity of measures is characterized by non-negativity of the weighted Ricci curvature, the inequality (1.5) is also established on weighted Riemannian manifolds with nonnegative weighted Ricci curvature as Klartag mentioned in [17, p.65] as follows.
Ric N is the weighted Ricci curvature which is defined in section 2. Note that Theorem 1.1 recovers the dilation inequality for s-concave probability measures in the Euclidean setting for s ∈ (−∞, 1/n]. Indeed, in virtue of the characterization of the s-concavity by Borell [8], we see that every s-concave probability measure on R n for s ≤ 1/n satisfies Ric 1/s ≥ 0 on its support (when s = 0, we put 1/s := ∞). More generally, the lower curvature bound Ric N ≥ K on weighted Riemannian manifolds is known to be equivalent to the curvature-dimension condition in the sense of Lott-Sturm-Villani (see [29], [31], [26], [27]).
The main purpose of this paper is to establish the sharp dilation type inequalities under more general curvature conditions, namely Ric N ≥ K for some K ∈ R. In our setting, we consider the dilation inequality (1.8) as an isoperimetric type inequality. Now, we introduce the dilation profile. For every ε ∈ [0, 1) and θ ∈ [0, 1], we define the ε-dilation profile of (M, g, m) by For instance, considering (R, | · |, µ s ) with s ∈ (−∞, 1/n] where µ s is the s-concave probability measure defined by (1.6), since it is the extremal of (1.5), we see that where we set 0 α := 1 for α > 0 by convention. When s = 0, the right hand side of (1.9) is interpreted In this paper, in addition to the dilation profile associated with ε ∈ [0, 1), we also treat the following dilation profile: for any Borel subset A ⊂ M, we define the dilation area of A by holds for any θ ∈ [0, 1]. Here, when N = ∞, the right hand side above is interpreted as −(1 − θ) log(1 − θ). Note that the dilation profile differs from the isoperimetric profile. In fact, the dilation profile is scale invariant, namely D (M,λ 2 g,m) = D (M,g,m) for any λ > 0 since the ε-dilation is also scale invariant, while the isoperimetric profile I (M,g,m) (for instance, see [23], [24] for the precise definition and overviews) satisfies I (M,λ 2 g,m) = I (M,g,m) /λ for any λ > 0. In order to describe our results, we introduce the following notations: given a nonzero integrable function f : We also denote for all κ, t ∈ R, and for all H, K, t ∈ R and N ∈ (−∞, ∞], where δ := K/(N − 1). Now, we also define the Curvature-Dimension-Diameter (CDD) dilation profile by We say that A ⊂ M is strongly-convex if for any p, q ∈ A, there exists a unique minimizing geodesic connecting p and q and is included in A. We also define the diameter of M by diamM := sup{d g (x, y) | x, y ∈ M}, where d g is the distance function canonically induced by g. In some special cases, the CDD dilation profiles have more concrete representations.
Case 7. If N ∈ (−∞, 0] and K > 0, In the above corollary, the infimums are considered in the pointwise sense. We can also observe that D K,N,D coincides with D K,N,D for a triple (K, N, D) in Corollary 1.3. Note also that the range of a triple (K, N, D) discussed in Corollary 1.3 is derived from the continuity in H ∈ R of entries in D K,N,D (see the proof of Corollary 1.3 and Remark 3.8). We note some remarks of Corollary 1.3. In general, when (M, g, m) satisfies the Curvature-Dimension-Diameter (CDD) condition, namely Ric N ≥ K and diamM ≤ D, then for any λ > 0, (M, λ 2 g, m) satisfies Ric N ≥ K/λ 2 and diamM ≤ λD. Thus, since the dilation area is invariant under scaling, we see that D K,N,∞ in Cases 1 and 4 coincides with D 1,N,∞ and that D 0,N,D in Cases 3, 5 and 8 are independent of D ∈ (0, ∞]. In particular, we emphasize that D K,N,∞ does not converge to D 0,N,∞ as K → 0. This paper is organized as follows. In section 2, we introduce the weighted Ricci curvature and the localization on a weighted Riemannian manifold. In section 3, we discuss the dilation inequality on R. In the first subsection, we give sufficient conditions such that the infimum of the dilation profile is attained at an interval. As its corollary, we obtain an explicit representation of the Gaussian dilation profile on R. In the next subsection, we complete the proofs of Theorem 1.2 and Corollary 1.3. In section 4, we furthermore discuss the dilation inequality associated with ε. In the final section, we prove a new type of functional inequalities related to entropies, derived from the comparison of the dilation profiles (1.10).

Acknowledgments
The author would like to thank Professor Shin-ichi Ohta for helpful comments and supports. This work was supported by JST, ACT-X Grant Number JPMJAX200J, Japan.

Preliminaries for weighted Riemannian manifolds 2.1 Localization associated with lower weighted Ricci curvature bounds
In this subsection, we introduce some notions on weighted Riemannian manifolds and the needle decomposition (also called the localization) constructed by Klartag in [17]. Using this decomposition, we can reduce our problem to the 1-dimensional one.
Let (M, g, m) be a geodesically-convex (namely, every two points can be connected by a minimizing geodesic) n-dimensional weighted Riemannian manifold with m = e −Ψ vol g and Ψ ∈ C ∞ (M), where vol g is the canonical Riemannian volume on M induced by g. For N ∈ (−∞, ∞], the weighted Ricci curvature Ric N is defined as for any p ∈ M and v ∈ T p M, where Ric g is the Ricci curvature on M canonically induced by g. We say that (M, g, m) satisfies the Curvature-Dimension (CD) condition CD(K, N ), or Ric N ≥ K, for some for any N ∈ (n, ∞) and N ′ ∈ (−∞, 1). The following theorem is the needle decomposition proved by Klartag [17] on a weighted Riemannian manifold, which has a lot of geometric and analytic applications (for instance [20], [28], [21]), and its extensions and applications in more general spaces are also investigated (see [11], [27]). (ii) For ν-almost every I ∈ Q, I is a minimizing geodesic in M, and µ I is supported on I. Moreover if I is not a singleton, then the density of µ I is smooth, and (I, | · |, µ I ) satisfies CD(K, N ).
In virtue of this localization, we can reduce our main assertion to the 1-dimensional one. Thus, we will discuss the dilation inequality on R in the next section. Now, we also recall that the ε-dilation A ε of a Borel subset A ⊂ M is defined by (1.7). Note that A ε is also a Borel subset. Indeed, setting c(ξ) := sup{t ≥ 0 | exp p (sv) is a minimizing geodesic for s ∈ [0, t] in M} > 0 for every ξ = (p, v) ∈ T M and X := {ξ ∈ T M | g(ξ, ξ) < c(ξ)}, we have, for any minimizing geodesic γ ξ (s) = exp p (sv) with s ∈ [0, 1] and ξ = (p, v) ∈ X, Since φ A is Borel measurable on X and c is continuous, the set X ε := {ξ ∈ X | φ A (ξ) > 1 − ε} is a Borel set, and hence A ε = A ∪ π(X ε ) is also a Borel set, where π : T M → M is the canonical projection.

(K, N)-convex functions
Let (M, g) be a geodesically-convex n-dimensional Riemannian manifold. For K ∈ R and N ∈ R\{0}, we say that a function ψ ∈ for any v ∈ T M. According to [12] for N > 0 and [26] for N < 0, there exists an equivalent representation as follows.
In particular, when a (probability) measure µ supported on an open interval I ⊂ R with a smooth density e −ψ(x) satisfies Ric N ≥ K for some K ∈ R and N ∈ (−∞, 1)∪(1, ∞), then ψ is (K, N −1)-convex. Thus, for any x ∈ I and t ∈ R with x + t ∈ I, Lemma 2.2 implies

Estimates for dilation areas
In this section, we discuss the dilation profile of a 1-dimensional weighted Riemannian manifold (I, | · |, µ) with µ(I) = 1, where I is an open interval in R. For simplicity, we denote D (I,|·|,µ) by D µ .
Before discussing the dilation inequality, we remark on the ε-dilation on R. In general, given proper subsets A ⊂ I ⊂ R and ε ∈ (0, 1), the ε-dilation of A in I, denoted by A 1 ε , does not coincide with the one in R, denoted by A 2 ε , since the former is necessarily included in I. However, we can observe that A 1 ε = A 2 ε ∩ I, and since we consider only a form µ(A ε ) = µ(A ε ∩ I) in our discussions, where µ is a (probability) measure supported on I, we consider the ε-dilation only in R even if the support of a discussed measure is not the whole space. The same problem occurs in more general spaces.

Existence and properties of minimizer on the real line
In this subsection, we consider sufficient conditions for a probability measure on R whose minimizer attaining the infimum of the dilation profile is an interval. In particular, our conditions will be satisfied by the Gaussian measures.
Proposition 3.1. Let µ be a probability measure on R whose density is e −ψ with ψ ∈ C 1 (R). Assume that there exists some ξ ∈ R such that ψ is non-increasing on (−∞, ξ] and non-decreasing on [ξ, ∞). In addition, we assume that for any x, y ∈ R with x < y, ψ(x) ≤ ψ(y) yields that Proof. Since the assertion is clear when θ = 0 and 1, we may assume that θ ∈ (0, 1). For fixed θ ∈ (0, 1), we will prove that for any interval A with µ(A) = θ, µ * (A) ≥ µ * (A θ ) holds. Without loss of generality, we may assume that A is open. Let A = (a, b) and take ε ∈ (0, 1). By the definition of the dilation, we obtain and hence
An important example is the Gaussian measures. Let ψ(x) := Kx 2 /2 + log 2π/K for some K > 0. Then for any x, y ∈ R with x < y and x 2 ≤ y 2 , Thus the Gaussian measures satisfy (3.1). More generally, if ψ is symmetric centered at ξ ∈ R in C 2 (R) and non-decreasing on [ξ, ∞) and ψ ′′ is non-increasing on [ξ, ∞), then ψ satisfies (3.1). Indeed, for x, y ∈ R with x < y and ψ(x) ≤ ψ(y), we obtain Therefore it follows from the mean-value theorem and the monotonicity of ψ ′′ that φ ′ is nonnegative for Furthermore, we note that any probability measure µ on [0, ∞) whose density f supported on [0, ∞) is non-increasing and satisfies (3 This assertion is also confirmed by the same argument as in Proposition 3.1. For instance, the probability measure µ s defined by (1.6) for s ≤ 0 satisfies these properties (although the same result holds for µ s with s ∈ (0, 1], we need additional (but not difficult) discussions since the support of µ s is compact). On the other hand, all log-concave probability measures on [0, ∞) with non-increasing densities do not satisfy (3.1). Indeed, we see that the probability measure whose density is proportional to e −x 3 1 [0,∞) (x) does not satisfy (3.1).
In particular, when a probability measure µ on R is centrally symmetric, then Theorem 3.2 implies 1], where e −ψ is the density of µ. The following proposition gives a sufficient condition such that µ * (A θ ) defined in Proposition 3.1 is concave on [0, 1].
where α(θ) ≥ 0 is given by for every θ ∈ [0, 1]. Since the differentiation in θ of (3.6) yields that Hence, the concavity of F on [0, 1] is equivalent to the non-increasing property of the function on the support of f . Since we see that by the log-concavity and non-increasing property of f on its support, we obtain Φ ′ ≤ 0 on the support of f . Hence Φ is non-increasing on the support of f , and we obtain the desired assertion.
As a corollary, we can give an explicit representation of the Gaussian dilation profile on R.
Corollary 3.4. The infimum of the dilation profile of the standard Gaussian measure is attained at a centrally symmetric interval. In particular, we have for every θ ∈ [0, 1], where γ 1 is the standard Gaussian measure on R and α(θ) ∈ [0, ∞] is given by 3.2 Proof of Theorem 1.2 In this subsection, we complete the proofs of Theorem 1.2 and Corollary 1.3. Proof. Fix θ ∈ [0, 1] and let A ⊂ I be an interval with µ(A) = θ. Since we easily see that D K,N,D (0) = D K,N,D (1) = 0, we may consider the assertion only for θ ∈ (0, 1). We will show that µ * (A) ≥ D K,N,D (θ). Let a, b ∈ R be the endpoints of A with a < b. By moving A left or right such that µ * (A) does not increase with keeping the volume θ as in Proposition 3.1, we may assume that A satisfies either ] means the boundary of supp(µ). Case 1. Suppose that A satisfies (3.8). By considering the reflection at the origin if necessary, we may also assume that ψ(b) ≥ ψ(a). Then (3.8) implies that −ψ ′ (b) ≤ ψ ′ (a). Since ψ is a (K, N − 1)-convex function by Ric N ≥ K, Lemma 2.2 and its subsequent discussion yield that for any x, t ∈ R with x, x + t ∈ I. It follows from (3.3) and (3.9) that for ε ∈ (0, 1), On the other hand, it follows from (3.9) that we obtain Therefore, by (3.10) and (3.11), we have Without loss of generality, we may assume that On the other hand, Thus, inequalities (3.13) and (3.14) also yield (3.12). This completes the proof. Now, we can prove Theorem 1.2 by Theorem 3.5 and Theorem 2.1.
Then by Theorem 2.1 for f , we have a partition Q of M, a measure ν on Q and a family {µ I } I∈Q of probability measures on M satisfying (i), (ii) and (iii) in Theorem 2.1. In particular, (iii) means that µ I (A) = θ 0 for ν-almost every I ∈ Q. Note also that since A is strongly-convex and ν-almost every I ∈ Q is a minimizing geodesic, A ∩ I is an interval. Since ν-almost every I ∈ Q is open and (I, | · |, µ I ) satisfies Ric N ≥ K and diamI ≤ D, it follows from Theorem 3.5 that µ * Hence, we obtain the desired assertion.
Next, we prove Corollary 1.3. In order to prove this corollary, we need the following two lemmas.
Proof. Note that µ * ((a, c)) = f (c)(c−a) follows from direct calculations as in (3.4) (or Case 2 in Theorem 3.5). When f is non-decreasing, we easily see that , c)).
Similarly, when f is non-increasing, we have ((a, c)), and hence we obtain the desired claim. ∈ (a, b), then for any θ ∈ (0, 1), Proof. By translation, we may assume that a = 0. For every θ ∈ (0, 1), we have where α(x) ∈ (0, b) is given by Now, the differentiation of (3.16) in x yields that and hence we have where we also used (3.16) in the second equality. Thus, we see that the claim of D ♭ (f, [0, x])(θ) being non-decreasing in x is equivalent to for any x ∈ (0, b). We can deduce this inequality from the assumption. Indeed, the integration by parts and the non-decreasing property of f for fixed θ ∈ (0, 1) and a, b ∈ R with 0 < b − a < D, which is derived from (3.12). When H varies from −∞ to ∞, then the first term in the right hand side of (3.18) monotonically and continuously (including the value ∞) varies from ∞ to 0, and the second term also monotonically and continuously varies from 0 to ∞ (also see [24,Proposition 3.3]). Thus, there exists a unique point H θ ∈ R satisfying Therefore, we have (3.21) Thus the right hand side of (3.19) becomes where H θ satisfies Case 2. Suppose N = ∞, K = 0 and D < ∞. As in Case 1, (3.21) yields that the right hand side of (3.19) becomes where H θ satisfies and by (3.20), it holds that (3.23) If D = ∞, by the integrability of J H θ ,0,∞ , we see that H θ < 0. Hence we obtain where we used the scale invariance of D ♭ (e −λt , [0, ∞)) for any λ > 0. Next, suppose D < ∞. Informally, since the dilation area is scale invariant and (I, λ| · |, µ) satisfies Ric N ≥ 0 and |I| ≤ λD for any λ > 0 when (I, | · |, µ) satisfies Ric N ≥ 0 and |I| ≤ D, letting λ → ∞, we can deduce (the same argument is true in Cases 5-2 and 8-2 below). More precisely, it follows from (3.22) and (3.23) that we obtain N ∈ (1, ∞) and K > 0. In this case, we see that

Case 4. Suppose
Thus, the right hand sides of (3.19) and (3.20) become respectively, which imply that which is strictly decreasing in t ∈ (0, π/ √ δ). Thus, by Lemma 3.7, we have It is easy to confirm that D ♭ ((−t) N −1 , [x, 0]) is independent of x by the scale invariance, and hence it yields Case 5-2. Suppose N ∈ (1, ∞), K = 0 and D < ∞. Since we have J H θ ,0,N (t) = (1+H θ t/(N −1)) N −1 + , the right hand side of (3.19) becomes and it follows from (3.20) that By Lemma 3.6, it holds that We can also confirm that D ♭ (1, [0, D])(θ) = θ by direct calculations. Hence, we obtain that Note that the function f (t) : for any ξ < 0, which is strictly decreasing on t < 0 independently of ξ. Thus for any −∞ < y < z ≤ 0, it follows from scale transformation and Lemma 3.7 that Hence, we obtain Case 6. Suppose N ∈ (1, ∞), K < 0 and D < ∞. In this case, we see that When |H θ /(N − 1)| = √ −δ, the right hand side of (3.19) becomes and the right hand side of (3.20) becomes Therefore, combining these with the argument in Case 3 for |H θ /(N − 1)| = √ −δ , it follows from Lemma 3.6 that Case 7. Suppose N ∈ (−∞, 0] and K > 0. Then we have and we exclude the case H θ (N −1) √ −δ ≤ −1 when D = ∞. When D = ∞, by the same argument as in Case 6, we obtain Similarly, when D < ∞, it holds that by Lemma 3.6, , and H θ is necessarily negative by the integrability of J H θ ,0,N . Thus, the right hand side of (3.19) becomes and it follows from (3.20) that Therefore, we obtain We easily see that D ♭ (t N −1 , [x, ∞)) is independent of x > 0 by the scale invariance, and hence it yields Note that the function f (t) for any ξ > 0, which is strictly decreasing on t > 0 independently of ξ. Thus for any 0 < y < z < ∞, it follows from Lemma 3.7 and scale transformation that, puttingz := max{z, y + D}, Hence, we obtain by scale transformation Case 9. Suppose N ∈ (−∞, 0], K < 0 and D < π/ √ δ. In this case, we see that Thus, by the same argument as in Case 4, we obtain Hence, we obtain the desired assertion. (2) When N = 0 and K = 0, we see that D 0,0,D = 0. Indeed, this follows from J H,0,0 ≡ ∞ for all H ∈ R when D = ∞. When D < ∞, we can also reduce this claim via the same argument as the proof of Case 8-2 above. Hence we excluded this case from Case 8 in Corollary 1.3.
(4) We emphasize that when K = 0 and N ∈ (−∞, 0) ∪ (1, ∞], we can completely recover (1.10) for any geodesically-convex n-dimensional weighted Riemannian manifold. For this purpose, we need to prove Theorem 1.2 for any Borel subset. By the same argument as in Theorem 1.2 via the needle decomposition, we may consider only the 1-dimensional case. Since D 0,N,∞ (which coincides with the right hand side of (1.10)) is concave on [0, 1], we can eventually reduce the 1-dimensional problem for a Borel subset to the one for an interval. However, this assertion is exactly proved in Theorem 1.2. Finally, note that the above argument is also applied to other cases if D K,N,D is concave.

Estimates for ε-dilation sets under some regularities
In this section, we consider the dilation inequalities associated with ε ∈ (0, 1). Given K ∈ R, N ∈ we obtain J = I ′ /I = (log I) ′ , which yields that for any θ ∈ (0, 1), Combining this equality with (4.1), we obtain for any θ ∈ (0, 1), Therefore, we can determine the function f from J. For the dilation inequality associated with ε below, we use similar functions constructed above via Corollary 1.3. Now, given a triple (K, N, D), we denote In order to ensure the existence of (4.2), we assume the following regularities.
We again use the needle decomposition to prove Theorem 4.1. Thus, similarly to the proof of Theorem 1.2 via Theorem 2.1, we consider only the 1-dimensional problem of Theorem 4.1. In order to prove the 1-dimensional problem, we need the followings. Proof. The monotonicity of F K,N,D (F −1 K,N,D (θ)/(1 − ε)) in ε immediately follows from the monotonicity of F K,N,D . We also see that Note that (4.4) holds for any θ ∈ (0, 1). In order to prove our assertion, it suffices to prove that H = D K,N,D holds on [0, 1]. We see that (4.4) is equivalent to We also see that if t 0 ∈ Y , then we have t 0 + δ ∈ Y for small enough δ > 0 as follows. Fixed t 0 ∈ Y with t 0 < 1, we suppose t 0 + δ / ∈ Y for any small enough δ > 0. Then there exists some t 1 ∈ (t 0 , 1] such that D K,N,D > H or D K,N,D < H holds on (t 0 , t 1 ). Without loss of generality, we may assume that D K,N,D > H holds on (t 0 , t 1 ). Then where µ − and µ + are normalized probability measures of µ on I ∩ (−∞, ξ] and I ∩ [ξ, ∞), respectively (when ξ coincides with one of the endpoints of I, then we adopt µ(A ε ) as the right hand side above).
Proof. Since the assertion is clear when I \ A consists of one connected component, we may assume that I \ A consists of two connected components. Moreover, without loss of generality, we may also assume that A is closed. Let G : I → R be the function defined as and denote A by [a, b]. Clearly, we have G(a) = 0 and G(b) = θ/µ((−∞, b]) > θ. Since G is continuous, there exists some point ξ ∈ int(A) such that G(ξ) = θ. Since it follows from the definition of the dilation that A ε includes the union of (A ∩ (−∞, ξ]) ε ∩ (−∞, ξ] and (A ∩ [ξ, ∞)) ε ∩ [ξ, ∞) whose intersection consists of only the element ξ, we see that where we used the elementary inequality ( 2,3,4) in the second inequality. On the other hand, since G(ξ) = θ, we obtain µ − (A) = θ and, equivalently, This completes the proof. Now, we shall prove Theorem 4.1. It suffices to show the following theorem by the same argument as in Theorem 1.2. The method of the proof is derived from the isoperimetric inequality discussed by Bobkov and Houdré in [5,Theorem 2.1].

Functional inequalities related to dilation profiles
Some preceding investigations including Bobkov and Nazarov [6] and Fradelizi [13] also studied the large and small deviation inequalities associated with certain parameters for a Borel function on R n (more precisely, the modulus of regularity or the Remez function) via the ε-dilation inequalities, which are applied to establishing the Kahane-Khintchine type inequality. In virtue of Thereom 1.1, we can also see that the same inequalities hold under CD(0, N ) on a geodesically-convex n-dimensional weighted Riemannian manifold via the same arguments in the Euclidean setting. In this section, we consider a new type of functional inequalities derived from the dilation profiles under CD(0, N ) with N ∈ (−∞, −1) ∪ [n, ∞].

The case N = ∞
Let (M, g, m) be a geodesically-convex n-dimensional weighted Riemannian manifold. We first introduce the measured Remez function which is also used by Fradelizi [13] without a measure. Equivalently, it holds that for any ε ∈ (0, 1) and λ > 0, Every measured Remez function is non-decreasing and satisfies u f ≥ 1 on [1, ∞) and u f (1) = 1. In addition, we define u ′ f (1) by Note that a Borel function does not always have its measured Remez function. For instance, when m is the n-dimensional Lebesgue measure on R n , then the characteristic function 1 A of any open proper subset A ⊂ R n satisfies u 1A = ∞ on (1, ∞). We can also deduce that for any q, a > 0 and nonnegative Borel function f with the measured Remez function, u af q (s) = u f (s) q holds for every s ∈ [1, ∞), which follows from the definition of the measured Remez function. Moreover since u f is continuous at s = 1, we obtain u ′ af q (1) = qu ′ f (1).  ∞). We also see that in general, these functions do not coincide. For instance, letting (R 2 , · 2 2 , m) be a weighted Riemannian manifold with a positive density on R 2 and f : R 2 → [0, ∞) be the characteristic function on R 2 \ I where I is a closed segment in R 2 , we can deduce that u f ≡ 1, butū f ≡ ∞ on (1, ∞).
According to [13], all norms · on R n satisfy u · (s) ≤ 2s − 1. More generally, all vector-valued polynomials P of degree at most d ≥ 1, namely In general, if (M, g, m) satisfies Ric ∞ ≥ K for some K > 0, then it satisfies the logarithmic Sobolev inequality with the constant K. Under Ric ∞ ≥ 0, Theorem 5.3 yields the following logarithmic Sobolev type inequality.
Corollary 5.4. Let (M, g, m) be a geodesically-convex n-dimensional weighted Riemannian manifold satisfying m(M ) = 1 and Ric ∞ ≥ 0. We also assume that (M, g, m) satisfies the Poincaré inequality with a constant C > 0 in the sense that, for any locally Lipschitz function h : M → R, it holds that Let f : M → R be a locally Lipschitz function and set a := M f dm. We assume that |f − a| has the measured Remez function u |f −a| . Then we have Combining this inequality with Theorem 5.3 and u ′ |f −a| 2 (1) = 2u ′ |f −a| (1), we obtain and finally, the Poincaré inequality yields Note that all log-concave probability measures on R n (equivalently, Ric ∞ ≥ 0) satisfy the Poincaré inequality (for instance, see [1]). For the Poincaré inequality on a weighted Riemannian manifold with Ric ∞ ≥ 0, see [22]. In general, it is known that weighted Riemannian manifolds with Ric ∞ ≥ 0 do not always satisfy the logarithmic Sobolev inequality, and hence we need to add an appropriate assumption. For instance, the logarithmic Sobolev inequality under the Gaussian isoperimetric inequality is investigated in [1].
In order to show Theorem 5.3, we first prove the following proposition which is regarded as a weak co-area type formula on dilation areas. For simplicity, given a Borel function f : By the definition of the measured Remez function, we deduce that ε dt Note that f 0 = f and f ε ≤ f on M for any ε ∈ (0, 1). We also define a function Φ f : for any log-concave probability measure µ and norm · on R n , where C > 0 is an absolute constant. On the other hand, under the same notations, (5.4) yields that by u ′ · (1) ≤ 2, In particular, our inequality is meaningful when p and q are close to each other. Moreover, when µ 0 is a probability measure on [0, ∞) whose density with respect to the 1-dimensional Lebesgue measure is e −x , since we see that for any n ∈ N, the measured Remez function of the ℓ ∞ -norm · ∞ with respect to µ ⊗n 0 in [0, ∞) n satisfies u · ∞ (s) = s for every s ≥ 1 by the same discussions after Proposition 5.5, Corollary 5.7 yields More generally, the following reverse Hölder inequality for polynomials can be easily proved by combining Corollary 5.7 with the comments after Remark 5.2.
Corollary 5.8. Let Ω ⊂ R n be a convex open subset and µ be a log-concave probability measure supported on Ω. We also take a normed vector space (V, · ). Then for any vector-valued polynomial P of degree at most d ≥ 1 from Ω to V defined as (5.1) and 0 < p ≤ q < ∞, we have We close this subsection by describing the reverse Hölder inequality for the distance function on a weighted Riemannian manifold corresponding to (5.5). Let (M, g) be a geodesically-convex n-dimensional Riemannian manifold and d g be the distance function induced by g. Now, fix x 0 ∈ M and define f : M → R as f (x) := d g (x, x 0 ). Then we can deduce that f has the Remez function in the sense of Remark 5.2 withū f (s) ≤ 2s − 1 for every s ≥ 1 as follows. Denote by B(r) the open ball centered at x 0 with a radius r > 0. It suffices to prove that for any r > 0 and ε ∈ (0, 1). First, note that given different two points x, y ∈ M, letting γ xy : [0, 1] → M be a minimizing geodesic from x to y, the triangle inequality yields f (γ xy (t)) ≥ |(1 − t)f (x) − tf (y)|. Now, fix r > 0 and ε ∈ (0, 1), and take x ∈ B(r) ε . By the definition of the ε-dilation, we can take y ∈ M such that |B(r) ∩ γ xy | > 1 − ε holds. In addition, we may assume that y belongs to B(r) by simple observations. Then we see that every t ∈ [0, 1] with γ xy (t) ∈ B(r) satisfies r ≥ f (γ xy (t)) ≥ |(1 − t)f (x) − tf (y)| ≥ (1 − t)f (x) − tf (y), and hence we obtain Since we have |B(r) ∩ γ xy | > 1 − ε, it yields which implies x ∈ B((1 + ε)r/(1 − ε)). Hence, we obtain (5.6).
The following corollary is the reverse Hölder inequality for the distance function.
The following theorem corresponds to Theorem 5.3.
In order to prove Theorem 5.11, we introduce the dual formula of the N -entropy that we postpone proving to the end of this subsection.  On the other hand, letting g = ρ 1/N (which implies g 1+N = ρ (1+N )/N ∈ L 1 (m)) yields equality in the above inequality. Consequently, we obtain Therefore, we obtain the reverse Young type inequality xy ≥ N 1 + N x (1+N )/N + 1 1 + N y 1+N , x ≥ 0, y > 0.