A (one-dimensional) free Brunn–Minkowski inequality

Ledoux, Michel

doi:10.1016/j.crma.2004.12.017

Cited by 13 publications

(22 citation statements)

References 16 publications

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“…[12]). As discussed in [24], this inequality may be used to deduce in an easy way both the Log-Sobolev and transportation inequalities.…”

Section: Brunn-minkowski Inequalitymentioning

confidence: 97%

“…Later, the first author [24] gave a simpler proof of (1.1) and (1.2) based on a free version of the geometric Brunn-Minkowski inequality obtained as a random matrix limiting case of its classical counterpart. He also showed the free analog of the Otto-Villani theorem indicating that the free Log-Sobolev inequality implies the free transportation inequality (1.2).…”

Section: (T (ρ))mentioning

confidence: 98%

“…The (one-dimensional) free Brunn-Minkowski inequality was put forward in [24] again through random matrix approximation. We provide here a direct mass transportation proof similar to the one of its classical (one-dimensional) counterpart (see e.g.…”

Section: Brunn-minkowski Inequalitymentioning

confidence: 99%

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Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

Ledoux

Popescu

2009

Journal of Functional Analysis

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This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on R + with the reference example of the Marcenko-Pastur distribution.

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“…[12]). As discussed in [24], this inequality may be used to deduce in an easy way both the Log-Sobolev and transportation inequalities.…”

Section: Brunn-minkowski Inequalitymentioning

confidence: 97%

Section: (T (ρ))mentioning

confidence: 98%

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Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

Ledoux

Popescu

2009

Journal of Functional Analysis

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“…In this section we show how the Log-Sobolev implies the transportation inequality. We use here the main idea from [17].…”

Section: Corollarymentioning

confidence: 99%

Free functional inequalities on the circle

Popescu

2018

Advances in Mathematics

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In this paper we deal with free functional inequalities on the circle. There are some interesting changes from their classical counterparts. For example, the free Poincaré inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities.These new phenomena also indicate that they have classical counterparts, which does not seem to have been investigated thus far. arXiv:1511.05274v3 [math.PR] 23 Oct 2017for any smooth function φ on the interval [−2, 2]. This is further refined in [7] for the case of the quadratic potential V (x) = x 2 /2. The purpose of this paper is to expand the study of free functional inequalities to the circle. Previously, such inequalities were first introduced by Hiai, Petz and Ueda in [14]. Particularly, the transportation and Log-Sobolev inequalities on the circle were proved using unitary random matrix ensembles. We should point out a key fact which will play an important role in the sequel. In the random matrix approach from [14], the key idea is to look at the Haar measure on U (n), the space of unitary matrices. The main issue with this is that U (n) has Ricci curvature constant in all directions, except one direction in which it is 0. Therefore, the Bakry-Émery condition which is typically used to get functional inequalities does not hold in this case. The fix is to actually look at the subgroup of matrices with determinant one. This has the Ricci curvature constant and thus, we can apply the classical inequalities, which in the limit provides the free analogs. This phenomena produces various corrections as we will discuss below.Our approach is not based on the random matrix approach, but rather on direct methods based on mass transport tools. We recapture the free transportation and Log-Sobolev alongside with HWI and Brunn-Minkowski. We also introduce the free Poincaré inequality.We start the discussion on our approach introducing first the free Poincaré inequality (which to our knowledge is new) similar to (1.4). If µ is a probability on the unit circle, we say it satisfies a free Poincaré inequality with constant ρ > 0 ifholds for any smooth function f on the circle. Here we denote by α the Haar measure on the circle. Notice that there are similarities and also differences to the classical analog and the free version (1.4). A difference from the classical counterpart is due to the fact that the left hand sides of (1.4) and (1.5) are independent of the measure µ. From a random matrix perspective the fact that the left hand side of (1.4) and (1.5) do not depend on the measure µ is a reflection of universality in random matrix theory for the fluctuations (see the argument in [18] where (1.4) is introduced). The other particularity of (1.5) is that as opposed to the (1.4) and the classical Poincaré inequality, the right hand side has an extra term, namely f dµ 2 . In the case of...

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“…Recently, M. Ledoux [17] used a similar random matrix technique to prove the free analogue of the Brunn-Minkowski inequality for measures on R, from which (together with the Hamilton-Jacobi approach) he gave short proofs of the free TCI and LSI for measures on R. Furthermore, his approach was shown in [10] to be still applicable for getting the free TCI in [12] for measures on T.…”

Section: Introductionmentioning

confidence: 99%