We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body Ω ⊂ ℝn, 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 Kμ. In particular, we obtain a simple proof of a conjecture of Kannan–Lovász–Simonovits for unit-balls of ℓnp, originally due to Sodin and Latała–Wojtaszczyk.