## The KLS isoperimetric conjecture for generalized Orlicz balls.(English)Zbl 1433.60005

The paper under review establishes the Kannan-Lovasz-Simonovits (KLS) isoperimetric conjecture for generalized Orlicz balls. Given a separable metric space $$(X, d)$$ with a Borel probability measure $$\mu$$, the Cheeger constant is defined by $D_{\mathrm{Che}}(X, d, \mu)= \inf_{A\subset X} \frac{\mu^+(A)}{\min(\mu (A), 1-\mu (A))},$ for all Borel sets $$A$$. Denote $$D_{\mathrm{Che}}(\mu) = D_{\mathrm{Che}}(\mathbb{R}^n, |\cdot |, \mu)$$ for Euclidean metric $$|\cdot |$$ in $$X=\mathbb{R}^n$$, and $$D_{\mathrm{Che}}^{\mathrm{Lin}}(\mu)$$ for only half space $$A=H \subset \mathbb{R}^n$$ in $$D_{\mathrm{Che}}(\mu)$$. Hence, $$D_{\mathrm{Che}}(\mu) \le D_{\mathrm{Che}}^{\mathrm{Lin}}(\mu)$$ in general. If $$\mu=\lambda_K$$ is a uniform Lebesgue probability measure on a convex compact set with nonempty interiors, Kannan, Lovasz and Simonovits [R. Kannan et al., Discrete Comput. Geom. 13, No. 3–4, 541–559 (1995; Zbl 0824.52012)] conjectured that $c D_{\mathrm{Che}}^{\mathrm{Lin}}(\lambda_K) \le D_{\mathrm{Che}}(\lambda_K),$ for some universal numeric constant $$c>0$$ independent of any other parameter such as $$n$$ or $$K$$. The authors KLS conjectured an essential equivalence between the former nonlinear isoperimetric inequality and its latter linear relaxation, it has been shown over the last two decades to be of fundamental importance to the understanding of volumetric and spectral properties of convex domains, revealing numerous connections to other central conjectures on the concentration of volume in convex bodies.
For the Poincaré inequality $$\|f - \int f d\mu\|_{L^2(\mu)} \le D_{\mathrm{Poin}}(\mu) \|\nabla f \|_{L^2(\mu)}$$, we have $$D_{\mathrm{Poin}}(\lambda_K) = \frac{1}{\sqrt{\lambda_1(K)}}$$ for the first non-zero eigenvalue of the Neumann Laplacian on $$K$$, and for only linear functions $$f$$ clearly $$D_{\mathrm{Poin}}^{\mathrm{Lin}}(\mu) \le D_{\mathrm{Poin}}(\mu)$$. V. G. Maz’ya [Sov. Math., Dokl. 1, 882–885 (1960; Zbl 0114.31001); translation from Dokl. Akad. Nauk SSSR 133, 527–530 (1960)], J. Cheeger [in: Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 195–199 (1970; Zbl 0212.44903)], P. Buser [Ann. Sci. Éc. Norm. Supér. (4) 15, 213–230 (1982; Zbl 0501.53030)] and M. Ledoux [Surv. Differ. Geom. 9, 219–240 (2004; Zbl 1061.58028)] showed that for all log-concave probability measures $$\mu$$ on $$\mathbb{R}^n$$, $\frac{1}{2} D_{\mathrm{Che}}(\mu) \le \frac{1}{D_{\mathrm{Poin}}(\mu)} \le C D_{\mathrm{Che}}(\mu),$ for some universal constant $$C > 1/2$$; the same inequality also holds for the corresponding linear relaxations $$D_{\mathrm{Che}}^{\mathrm{Lin}}(\mu)$$ and $$D_{\mathrm{Poin}}^{\mathrm{Lin}}(\mu)$$. Consequently, the KLS conjecture may be equivalently reformulated as the following, $D_{\mathrm{Poin}}(\mu) \le C D_{\mathrm{Poin}}^{\mathrm{Lin}}(\mu),$ for some universal constant $$C>1$$ and all log-concave probability measures $$\mu$$ on $$\mathbb{R}^n$$.
Section 1 starts by explaining the KLS conjecture and previous known results, as well as a generalized Orlicz balls in $$\mathbb{R}^n$$ ($$K_E = \{x\in \mathbb{R}^n: \sum_{i=1}^n V_i(x_i) \le E\}$$ for $$n$$ one-dimensional convex functions $$V_i: \mathbb{R}\to \mathbb{R}$$). Theorem 1.1 states the simplified main theorem for this paper, for $$\mu_i = \exp (- V_i (y))dy$$ a log-concave probability measure and a random-variable $$X_i$$ distributed according to $$\mu_i$$ with $$E_V = 1 + \sum_{i=1}^n EV_i(X_i)$$, then $$E=E_V\le n+1$$ and $$C^{-n}\le \mathrm{Vol}(K_E) \le C^n$$, $D_{\mathrm{Poin}}(\lambda_{K_E}) \le C \log (e + A^{(2)} \wedge n) D_{\mathrm{Poin}}^{\mathrm{Lin}}(\lambda_{K_E}).$ This confirms the KLS conjecture for the generalized Orlicz ball $$K_E$$. Theorem 1.2 further shows that the KLS conjecture is true for $$K_E= \{x\in \mathbb{R}^n: V(x) \le E\}$$ and $$A^{(2)} \wedge n =\sqrt{n} D_{\mathrm{Poin}}(\mu)$$. Various examples are explicitly presented.
Section 2 states results from Theorem 2.1 (main technical theorem) to Theorem 2.4 (main theorem) with technical supports from Corollary 2.2 and Proposition 2.3. Section 3 introduces concentration profiles and Barthe-Milman’s result (Proposition 3.1) on related concentrations $${K}_1$$ and $${K}_2$$. For the concentration inequality for sums of independent random variables bounded from only one side (Hoeffding inequality for bounded random variables), Theorem 3.6 shows the one-sided Hoeffding inequality in the spirit of Bernstein’s inequality. Section 4 gives the volume $$\mathrm{Vol}(K_E)$$ bounding it first in Proposition 4.1. Part of Proposition 2.3 is proved in Proposition 4.6 as refinement of Fradelizi’s bound which is sharp whenever $$\mu$$ is log-affine on an appropriate convex cone, and is also proved in Proposition 4.9 for the barycenter and the covariance matrix of $$K_E$$. Section 5 initiates the 1-Wasserstein distance of the induced probability measure on the boundary of a star-shaped body and a probability measure from the Borel function with indicator bound first, then an $$L^1$$-version of Hardy type inequality is proven by reducing various spectral-gap question from the star-shaped body to its boundary $$\partial \Omega$$ in Lemma 5.4.
Section 6 completes the proof of main technical theorem and to puts everything together. Subsection 6.1 first presents the proof of Theorem 2.1 by reducing the condition on $$D_{\mathrm{Poin}}(\nu)^2/D_{\mathrm{Poin}}^{\mathrm{Lin}}(\nu)^2 \in [1, 12]$$ to have the KLS conjecture valid. By a well-known result of M. Gromov and V. D. Milman [Am. J. Math. 105, 843–854 (1983; Zbl 0522.53039)], a Poincaré inequality always implies the following exponential concentration: $$K_{\nu}(r) \le e^{- c_0 \frac{r}{D_{\mathrm{Poin}}(\nu)}}$$. By applying Proposition 3.5 and Proposition 4.1, The concentration $$K_{\mu_{K_E}, w_0/\sqrt{n}}(r)$$ has a exponential bound and the $$L^1$$-version of the Hardy type inequality estimates. It remains to invoke the following result, established in a more general weighted Riemannian setting, asserting the equivalence between concentration, spectral-gap and linear-isoperimetry under appropriate convexity assumptions. Theorem 2.1 follows. Subsection 6.2 presents a proof of Theorem 2.5, and the proof of Theorem 2.5 is identical to the one of Theorem 2.1 described in subsection 6.1, with the only difference being in the first step – instead of invoking the $$L^p$$ estimate given by Proposition 3.5 for transferring concentration from $$\mu$$ to $$\mu_{K_E, w}$$ and to invoke the $$L^{\infty}$$ estimate of Lemma 3.4. Subsection 6.3 conclude the proofs of assertion (8) of Proposition 2.3 (by the well-known bath-tub principle), Theorem 2.4 (by Proposition 2.3) and Theorems 1.1 (The dimension-independent part immediately follows from an application of Theorem 2.4 for any $$E\in [E_{min}, E_{max}]$$) and Theorem 1.2 (by Theorem 2.5 to $$\mu = \mu_1 \otimes \mu_2 \cdots \otimes \mu_n$$). Subsection 6.4 gives a general formulation after rescaling, and states the main theorem (generalized version) in Corollary 6.3, and subsection 6.5 establishes the previous Examples 1.4 and 1.5.
For convex bounded domains other than the generalized Orlicz balls or other than log-concave measures, the KLS conjecture remains open.

### MSC:

 60D05 Geometric probability and stochastic geometry 52A23 Asymptotic theory of convex bodies 46B07 Local theory of Banach spaces
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### References:

 [1] Alonso-Gutierrez, D. and Bastero, J. (2015). Approaching the Kannan–Lovasz–Simonovits and Variance Conjectures. Lecture Notes in Math.2131. Springer, Berlin. · Zbl 1318.52001 [2] Anttila, M., Ball, K. and Perissinaki, I. (2003). The central limit problem for convex bodies. Trans. Amer. Math. Soc.355 4723–4735. · Zbl 1033.52003 [3] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math.1123 177–206. Springer, Berlin. · Zbl 0561.60080 [4] Ball, K. and Nguyen, V. H. (2012). Entropy jumps for isotropic log-concave random vectors and spectral gap. Studia Math.213 81–96. · Zbl 1264.94077 [5] Barthe, F. and Cordero-Erausquin, D. (2013). Invariances in variance estimates. Proc. Lond. Math. Soc. (3) 106 33–64. · Zbl 1281.60020 [6] Barthe, F. and Milman, E. (2013). Transference principles for log-Sobolev and spectral-gap with applications to conservative spin systems. Comm. Math. Phys.323 575–625. · Zbl 1297.82011 [7] Barthe, F. and Wolff, P. (2009). Remarks on non-interacting conservative spin systems: The case of gamma distributions. Stochastic Process. Appl.119 2711–2723. · Zbl 1169.60325 [8] Bentkus, V. (2004). On Hoeffding’s inequalities. Ann. Probab.32 1650–1673. · Zbl 1062.60011 [9] Bobkov, S. (1996). Extremal properties of half-spaces for log-concave distributions. Ann. Probab.24 35–48. · Zbl 0859.60048 [10] Bobkov, S. (2007). On isoperimetric constants for log-concave probability distributions. In Geometric Aspects of Functional Analysis, Israel Seminar 2004–2005. Lecture Notes in Math.1910 81–88. Springer, Berlin. · Zbl 1132.60301 [11] 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 [12] Bobkov, S. G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab.27 1903–1921. · Zbl 0964.60013 [13] Bobkov, S. G. (2003). Spectral gap and concentration for some spherically symmetric probability measures. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.1807 37–43. Springer, Berlin. · Zbl 1052.60003 [14] Bobkov, S. G. and Houdré, C. (1997). Isoperimetric constants for product probability measures. Ann. Probab.25 184–205. · Zbl 0878.60013 [15] Bobkov, S. G. and Koldobsky, A. (2003). On the central limit property of convex bodies. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.1807 44–52. Springer, Berlin. · Zbl 1039.52003 [16] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat.12 239–252. · Zbl 0297.60004 [17] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford. · Zbl 1279.60005 [18] Brazitikos, S., Giannopoulos, A., Valettas, P. and Vritsiou, B.-H. (2014). Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs196. Amer. Math. Soc., Providence, RI. · Zbl 1304.52001 [19] Buser, P. (1982). A note on the isoperimetric constant. Ann. Sci. Éc. Norm. Supér. (4) 15 213–230. · Zbl 0501.53030 [20] Caffarelli, L. A. (2000). Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys.214 547–563. · Zbl 0978.60107 [21] Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969) 195–199. Princeton Univ. Press, Princeton, NJ. [22] Eldan, R. (2013). Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal.23 532–569. · Zbl 1277.52013 [23] Eldan, R. and Klartag, B. (2011). Approximately Gaussian marginals and the hyperplane conjecture. In Concentration, Functional Inequalities and Isoperimetry (C. Houdré, M. Ledoux, E. Milman and M. Milman, eds.). Contemporary Mathematics545 55–68. Amer. Math. Soc. · Zbl 1235.52012 [24] Fleury, B. (2010). Between Paouris concentration inequality and variance conjecture. Ann. Inst. Henri Poincaré Probab. Stat.46 299–312. · Zbl 1214.46006 [25] Fleury, B. (2010). Concentration in a thin Euclidean shell for log-concave measures. J. Funct. Anal.259 832–841. · Zbl 1200.60013 [26] Fleury, B. (2012). Poincaré inequality in mean value for Gaussian polytopes. Probab. Theory Related Fields152 141–178. · Zbl 1236.52007 [27] Fradelizi, M. (1997). Sections of convex bodies through their centroid. Arch. Math. (Basel) 69 515–522. · Zbl 0901.52005 [28] Fradelizi, M. (1999). Hyperplane sections of convex bodies in isotropic position. Beitr. Algebra Geom.40 163–183. · Zbl 0966.52004 [29] Gardner, R. J. (2002). The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39 355–405. · Zbl 1019.26008 [30] Gromov, M. and Milman, V. D. (1983). A topological application of the isoperimetric inequality. Amer. J. Math.105 843–854. · Zbl 0522.53039 [31] Grünbaum, B. (1960). Partitions of mass-distributions and of convex bodies by hyperplanes. Pacific J. Math.10 1257–1261. · Zbl 0101.14603 [32] Guédon, O. and Milman, E. (2011). Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal.21 1043–1068. · Zbl 1242.60012 [33] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc.58 13–30. · Zbl 0127.10602 [34] Huet, N. (2011). Spectral gap for some invariant log-concave probability measures. Mathematika57 51–62. · Zbl 1219.28004 [35] Kannan, R., Lovász, L. and Simonovits, M. (1995). Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom.13 541–559. · Zbl 0824.52012 [36] Kim, Y. H. and Milman, E. (2012). A generalization of Caffarelli’s contraction theorem via (reverse) heat flow. Math. Ann.354 827–862. · Zbl 1257.35101 [37] Klartag, B. (2006). On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal.16 1274–1290. · Zbl 1113.52014 [38] Klartag, B. (2007). A central limit theorem for convex sets. Invent. Math.168 91–131. · Zbl 1144.60021 [39] Klartag, B. (2007). Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal.245 284–310. · Zbl 1140.52004 [40] Klartag, B. (2007). Uniform almost sub-Gaussian estimates for linear functionals on convex sets. Algebra i Analiz19 109–148. [41] Klartag, B. (2009). A Berry–Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields45 1–33. · Zbl 1171.60322 [42] Klartag, B. (2014). Concentration of measures supported on the cube. Israel J. Math.203 59–80. · Zbl 1357.60020 [43] Klartag, B. and Milman, V. D. (2005). Geometry of log-concave functions and measures. Geom. Dedicata112 169–182. · Zbl 1099.28002 [44] Kolesnikov, A. V. and Milman, E. (2014). Remarks on the KLS conjecture and Hardy-type inequalities. In Geometric Aspects of Functional Analysis, Israel Seminar 2011–2013. Lecture Notes in Math.2116 273–292. Springer, Berlin. · Zbl 1318.52008 [45] Kolesnikov, A. V. and Milman, E. (2016). Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities. Calc. Var. Partial Differential Equations55 Art. 77, 36 pp. · Zbl 1355.53031 [46] Krugova, E. P. (1995). Differentiability of convex measures. Mat. Zametki58 862–871, 960. Translation in Math. Notes58(5–6) (1995) 1294–1301 (1996). · Zbl 0856.46026 [47] Latała, R. and Wojtaszczyk, J. O. (2008). On the infimum convolution inequality. Studia Math.189 147–187. · Zbl 1161.26010 [48] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs89. Amer. Math. Soc., Providence, RI. · Zbl 0995.60002 [49] Ledoux, M. (2004). Spectral gap, logarithmic Sobolev constant, and geometric bounds. In Surveys in Differential Geometry. Vol. IX 219–240. International Press, Somerville, MA. · Zbl 1061.58028 [50] Lee, Y. T. and Vempala, S. (2017). Eldan’s stochastic localization and the KLS hyperplane conjecture: An improved lower bound for expansion. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) 998–1007. [51] Maurer, A. (2003). A bound on the deviation probability for sums of non-negative random variables. JIPAM. J. Inequal. Pure Appl. Math.4 Article 15, 6. · Zbl 1021.60036 [52] Maz’ja, V. G. (1960). Classes of domains and imbedding theorems for function spaces. Dokl. Akad. Nauk SSSR3 527–530. Engl. transl. Soviet Math. Dokl.1 (1961) 882–885. [53] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin.16 195–248. Springer, Berlin. · Zbl 0927.60027 [54] Milman, E. (2009). On the role of convexity in isoperimetry, spectral-gap and concentration. Invent. Math.177 1–43. · Zbl 1181.52008 [55] Milman, E. (2010). Isoperimetric and concentration inequalities—Equivalence under curvature lower bound. Duke Math. J.154 207–239. · Zbl 1205.53038 [56] Milman, E. (2011). Isoperimetric bounds on convex manifolds. In Concentration, Functional Inequalities and Isoperimetry (C. Houdré, M. Ledoux, E. Milman and M. Milman, eds.). Contemp. Math.545 195–208. Amer. Math. Soc. · Zbl 1229.60021 [57] Milman, E. (2012). Properties of isoperimetric, functional and transport-entropy inequalities via concentration. Probab. Theory Related Fields152 475–507. · Zbl 1239.60012 [58] Milman, E. (2017). Beyond traditional curvature-dimension I: New model spaces for isoperimetric and concentration inequalities in negative dimension. Trans. Amer. Math. Soc.369 3605–3637. · Zbl 1362.53047 [59] Milman, V. D. and Pajor, A. (1987–1988). Isotropic position and interia ellipsoids and zonoids of the unit ball of a normed $$n$$-dimensional space. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.1376 64–104. Springer, Berlin. [60] Morgan, F. (2009). Geometric Measure Theory (a Beginner’s Guide), 4th ed. Elsevier/Academic Press, Amsterdam. · Zbl 1179.49050 [61] Petrov, V. V. (1975). Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete82. Springer, New York. Translated from the Russian by A. A. Brown. · Zbl 0322.60042 [62] Pilipczuk, M. and Wojtaszczyk, J. O. (2008). The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball. Positivity12 421–474. · Zbl 1155.60006 [63] Schmuckenschläger, M. (1998). Martingales, Poincaré type inequalities, and deviation inequalities. J. Funct. Anal.155 303–323. [64] Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications44. Cambridge Univ. Press, Cambridge. [65] Sodin, S. (2008). An isoperimetric inequality on the $$ℓ_{p}$$ balls. Ann. Inst. Henri Poincaré B, Probab. Stat.44 362–373. · Zbl 1181.60025 [66] Sternberg, P. and Zumbrun, K. (1999). On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint. Comm. Anal. Geom.7 199–220. · Zbl 0930.49024 [67] Villani, C. (2009). Optimal Transport—Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin. · Zbl 1156.53003 [68] Wojtaszczyk, J. O. (2007). The square negative correlation property for generalized Orlicz balls. In Geometric Aspects of Functional Analysis. Lecture Notes in Math.1910 305–313. · Zbl 1137.60007
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