## Bernoulli

• Bernoulli
• Volume 25, Number 3 (2019), 1794-1815.

### On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one

#### Abstract

First, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Second, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger’s inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincaré inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincaré inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger’s inequality, which is a $L_{p}$-Poincaré inequality for $p=2$, to any real $p\geq1$.

#### Article information

Source
Bernoulli, Volume 25, Number 3 (2019), 1794-1815.

Dates
Revised: March 2018
First available in Project Euclid: 12 June 2019

https://projecteuclid.org/euclid.bj/1560326428

Digital Object Identifier
doi:10.3150/18-BEJ1036

Mathematical Reviews number (MathSciNet)
MR3961231

Zentralblatt MATH identifier
07066240

#### Citation

Saumard, Adrien; Wellner, Jon A. On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one. Bernoulli 25 (2019), no. 3, 1794--1815. doi:10.3150/18-BEJ1036. https://projecteuclid.org/euclid.bj/1560326428

#### References

• [1] Adamczak, R., Latała, R., Litvak, A.E., Oleszkiewicz, K., Pajor, A. and Tomczak-Jaegermann, N. (2014). A short proof of Paouris’ inequality. Canad. Math. Bull. 57 3–8.
• [2] Adamczak, R., Latała, R., Litvak, A.E., Pajor, A. and Tomczak-Jaegermann, N. (2014). Tail estimates for norms of sums of log-concave random vectors. Proc. Lond. Math. Soc. (3) 108 600–637.
• [3] 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. With a preface by Dominique Bakry and Michel Ledoux.
• [4] Block, H.W. and Fang, Z.B. (1988). A multivariate extension of Hoeffding’s lemma. Ann. Probab. 16 1803–1820.
• [5] Bobkov, S. and Ledoux, M. (2014). One-dimensional empirical measures, order statistics and Kantorovich transport distances. Preprint.
• [6] Bobkov, S.G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28.
• [7] Bobkov, S.G., Götze, F. and Houdré, C. (2001). On Gaussian and Bernoulli covariance representations. Bernoulli 7 439–451.
• [8] Bobkov, S.G. and Houdré, C. (1997). Isoperimetric constants for product probability measures. Ann. Probab. 25 184–205.
• [9] Bobkov, S.G. and Houdré, C. (1997). Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129 viii $+$ 111.
• [10] Bobkov, S.G. and Zegarlinski, B. (2005). Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176 x $+$ 69.
• [11] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239–252.
• [12] Bubeck, S. and Eldan, R. (2014). The entropic barrier: A simple and optimal universal self-concordant barrier. Preprint. Available at arXiv:1412.1587.
• [13] Carlen, E.A., Cordero-Erausquin, D. and Lieb, E.H. (2013). Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures. Ann. Inst. Henri Poincaré Probab. Stat. 49 1–12.
• [14] Fougères, P. (2005). Spectral gap for log-concave probability measures on the real line. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 95–123. Berlin: Springer.
• [15] Fradelizi, M. and Guédon, O. (2006). A generalized localization theorem and geometric inequalities for convex bodies. Adv. Math. 204 509–529.
• [16] Hoeffding, W. (1994). The Collected Works of Wassily Hoeffding. Springer Series in Statistics: Perspectives in Statistics. New York: Springer. Edited and with a preface by N.I. Fisher and P.K. Sen.
• [17] Höffding, W. (1940). Maszstabinvariante Korrelationstheorie. Schr. Math. Inst. U. Inst. Angew. Math. Univ. Berlin 5 181–233.
• [18] Houdré, C. (2002). Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 1223–1237.
• [19] Houdré, C. and Marchal, P. (2004). On the concentration of measure phenomenon for stable and related random vectors. Ann. Probab. 32 1496–1508.
• [20] Latała, R. (2011). Weak and strong moments of random vectors. In Marcinkiewicz Centenary Volume. Banach Center Publ. 95 115–121. Warsaw: Polish Acad. Sci. Inst. Math.
• [21] Latała, R. (2017). On some problems concerning log-concave random vectors. In Convexity and Concentration (E. Carlen, M. Madiman and E.M. Werner, eds.) 525–539. New York, NY: Springer.
• [22] Latała, R. and Strzelecka, M. (2016). Weak and strong moments of $\ell_{r}$-norms of log-concave vectors. Proc. Amer. Math. Soc. 144 3597–3608.
• [23] Latała, R. and Wojtaszczyk, J.O. (2008). On the infimum convolution inequality. Studia Math. 189 147–187.
• [24] Ledoux, M. (2004). Spectral gap, logarithmic Sobolev constant, and geometric bounds. In Surveys in Differential Geometry. Vol. IX. Surv. Differ. Geom. 9 219–240. Somerville, MA: Int. Press.
• [25] Menz, G. and Otto, F. (2013). Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Ann. Probab. 41 2182–2224.
• [26] Miclo, L. (1999). An example of application of discrete Hardy’s inequalities. Markov Process. Related Fields 5 319–330.
• [27] Milman, E. (2009). On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 1–43.
• [28] Paouris, G. (2006). Concentration of mass on convex bodies. Geom. Funct. Anal. 16 1021–1049.
• [29] Saumard, A. and Wellner, J.A. (2014). Log-concavity and strong log-concavity: A review. Stat. Surv. 8 45–114.
• [30] Saumard, A. and Wellner, J.A. (2017). On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one. Technical report, Department of Statistics, University of Washington. Available at arXiv:1711.00668.
• [31] Saumard, A. and Wellner, J.A. (2018). Efron’s monotonicity property for measures on $\mathbb{R}^{2}$. J. Multivariate Anal. 166 212–224.