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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 $Ω\subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial Ω$. This reduces the study of the Neumann Poincaré constant on $Ω$ to that of the cone and Lebesgue measures on $\partial Ω$; these may be bounded via the curvature of $\partial Ω$. 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 $\ell^n_p$, originally due to Sodin and Latała--Wojtaszczyk.