  We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary
functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily
vanishing on the boundary $\partial \Omega$. This reduces the study of the
Neumann Poincar\'e constant on $\Omega$ to that of the cone and Lebesgue
measures on $\partial \Omega$; these may be bounded via the curvature of
$\partial \Omega$. A second reduction is obtained to the class of harmonic
functions on $\Omega$. We also study the relation between the Poincar\'e
constant of a log-concave measure $\mu$ and its associated K. Ball body
$K_\mu$. In particular, we obtain a simple proof of a conjecture of
Kannan--Lov\'asz--Simonovits for unit-balls of $\ell^n_p$, originally due to
Sodin and Lata{\l}a--Wojtaszczyk.
