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Remarks on the KLS Conjecture and Hardy-Type Inequalities

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2116))

Abstract

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body \(\varOmega \subset \mathbb{R}^{n}\), not necessarily vanishing on the boundary ∂ Ω. This reduces the study of the Neumann Poincaré constant on Ω to that of the cone and Lebesgue measures on ∂ Ω; these may be bounded via the curvature of ∂ Ω. A second reduction is obtained to the class of harmonic functions on Ω. We also study the relation between the Poincaré constant of a log-concave measure μ and its associated K. Ball body K μ . In particular, we obtain a simple proof of a conjecture of Kannan–Lovász–Simonovits for unit-balls of ℓ p n, originally due to Sodin and Latała–Wojtaszczyk.

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Acknowledgements

We would like to thank Bo’az Klartag for his interest and fruitfull discussions. The first-named author was supported by RFBR project 12-01-33009 and the DFG project CRC 701. This study (research grant No 14-01-0056) was supported by The National Research University–Higher School of Economics’ Academic Fund Program in 2014/2015. The second-named author was supported by ISF (grant no. 900/10), BSF (grant no. 2010288), Marie-Curie Actions (grant no. PCIG10-GA-2011-304066) and the E. and J. Bishop Research Fund.

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Authors and Affiliations

  1. Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia

    Alexander V. Kolesnikov

  2. Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel

    Emanuel Milman

Authors
  1. Alexander V. Kolesnikov
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  2. Emanuel Milman
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Correspondence to Emanuel Milman .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Bo'az Klartag

  2. Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Kolesnikov, A.V., Milman, E. (2014). Remarks on the KLS Conjecture and Hardy-Type Inequalities. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_18

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