We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body Ω ⊂ R n , not necessarily vanishing on the boundary ∂Ω. This reduces the study of the Neumann Poincar´e 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 Poincar´e 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–Lov´asz–Simonovits for unit-balls of ℓ n p , originally due to Sodin and Lata la–Wojtaszczyk.