Remarks on the KLS conjecture and Hardy-type inequalities

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 Poincare 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 Poincare 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–Lovasz–Simonovits for unit-balls of l p n , originally due to Sodin and Latala–Wojtaszczyk.

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