# zbMATH — the first resource for mathematics

On the convex Poincaré inequality and weak transportation inequalities. (English) Zbl 1442.60030
Summary: We prove that for a probability measure on $$\mathbb{R}^{n}$$, the Poincaré inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan, Roberto, Samson, Shu, Tetali and Feldheim, Marsiglietti, Nayar, Wang, concerning probability measures on the real line.
The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest.
We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by S. G. Bobkov et al. [Lect. Notes Math. 2169, 25–53 (2017; Zbl 1366.60052)].

##### MSC:
 60E15 Inequalities; stochastic orderings 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 60D05 Geometric probability and stochastic geometry
Full Text:
##### References:
 [1] Adamczak, R., Bednorz, W. and Wolff, P. (2017). Moment estimates implied by modified log-Sobolev inequalities. ESAIM Probab. Stat.21 467–494. · Zbl 1393.60024 [2] Adamczak, R. and Strzelecki, M. (2015). Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions. Studia Math.230 59–93. · Zbl 1331.60036 [3] Adamczak, R. and Wolff, P. (2015). Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probab. Theory Related Fields162 531–586. · Zbl 1323.60033 [4] Aida, S. and Stroock, D. (1994). Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett.1 75–86. · Zbl 0862.60064 [5] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses [Panoramas and Syntheses] 10. Paris: Société Mathématique de France. [6] Bobkov, S. and Ledoux, M. (1997). Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields107 383–400. · Zbl 0878.60014 [7] Bobkov, S.G., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 80 669–696. · Zbl 1038.35020 [8] Bobkov, S.G. and Götze, F. (1999). Discrete isoperimetric and Poincaré-type inequalities. Probab. Theory Related Fields114 245–277. · Zbl 0940.60028 [9] Bobkov, S.G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal.163 1–28. · Zbl 0924.46027 [10] Bobkov, S.G., Nayar, P. and Tetali, P. (2017). Concentration properties of restricted measures with applications to non-Lipschitz functions. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.2169 25–53. Cham: Springer. · Zbl 1366.60052 [11] Chafaï, D. and Malrieu, F. (2010). On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities. Ann. Inst. Henri Poincaré Probab. Stat.46 72–96. · Zbl 1204.60025 [12] de la Peña, V.H., Klass, M.J. and Lai, T.L. (2004). Self-normalized processes: Exponential inequalities, moment bounds and iterated logarithm laws. Ann. Probab.32 1902–1933. · Zbl 1075.60014 [13] Evans, L.C. (2010). Partial Differential Equations, 2nd ed. Graduate Studies in Mathematics19. Providence, RI: Amer. Math. Soc. · Zbl 1194.35001 [14] Feldheim, N., Marsiglietti, A., Nayar, P. and Wang, J. (2018). A note on the convex infimum convolution inequality. Bernoulli24 257–270. · Zbl 1387.60035 [15] Gluskin, E.D. and Kwapień, S. (1995). Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math.114 303–309. · Zbl 0834.60050 [16] Gozlan, N., Roberto, C. and Samson, P.-M. (2014). Hamilton Jacobi equations on metric spaces and transport entropy inequalities. Rev. Mat. Iberoam.30 133–163. · Zbl 1296.60040 [17] Gozlan, N., Roberto, C. and Samson, P.M. (2015). From dimension free concentration to the Poincaré inequality. Calc. Var. Partial Differential Equations52 899–925. · Zbl 1352.60028 [18] Gozlan, N., Roberto, C., Samson, P.M., Shu, Y. and Tetali, P. (2017). Characterization of a class of weak transport-entropy inequalities on the line. Ann. Inst. Henri Poincaré Probab. Stat. To appear. Available at arXiv:1509.04202v2. · Zbl 1404.60030 [19] Gozlan, N., Roberto, C., Samson, P.-M. and Tetali, P. (2017). Kantorovich duality for general transport costs and applications. J. Funct. Anal.273 3327–3405. · Zbl 1406.60032 [20] Gromov, M. and Milman, V.D. (1983). A topological application of the isoperimetric inequality. Amer. J. Math.105 843–854. · Zbl 0522.53039 [21] Ledoux, M. (1995/97). On Talagrand’s deviation inequalities for product measures. ESAIM Probab. Stat.1 63–87 (electronic). · Zbl 0869.60013 [22] Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. In Séminaire de Probabilités, XXXIII. Lecture Notes in Math.1709 120–216. Berlin: Springer. · Zbl 0957.60016 [23] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs89. Providence, RI: Amer. Math. Soc. [24] Maurey, B. (1991). Some deviation inequalities. Geom. Funct. Anal.1 188–197. · Zbl 0756.60018 [25] Milman, E. (2009). On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math.177 1–43. · Zbl 1181.52008 [26] Paouris, G. and Valettas, P. (2017). A small deviation inequality for convex functions. Ann. Probab. To appear. Available at arXiv:1611.01723. · Zbl 1429.60022 [27] Pisier, G. (1986). Probabilistic methods in the geometry of Banach spaces. In Probability and Analysis (Varenna, 1985). Lecture Notes in Math.1206 167–241. Berlin: Springer. [28] Samson, P.M. (2000). Concentration of measure inequalities for Markov chains and $$Φ$$-mixing processes. Ann. Probab.28 416–461. · Zbl 1044.60061 [29] Samson, P.M. (2003). Concentration inequalities for convex functions on product spaces. In Stochastic Inequalities and Applications. Progress in Probability56 33–52. Basel: Birkhäuser. · Zbl 1037.60019 [30] Talagrand, M. (1991). A new isoperimetric inequality and the concentration of measure phenomenon. In Geometric Aspects of Functional Analysis (1989–90). Lecture Notes in Math.1469 94–124. Berlin: Springer. · Zbl 0818.46047 [31] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math.81 73–205. · Zbl 0864.60013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.