  Let $\mu$ be a probability measure on $\rr^n$ ($n \geq 2$) with Lebesgue
density proportional to $e^{-V (\Vert x\Vert )}$, where $V : \rr_+ \to \rr$ is
a smooth convex potential. We show that the associated spectral gap in $L^2
(\mu)$ lies between $(n-1) / \int_{\rr^n} \Vert x\Vert ^2 \mu(dx)$ and $n /
\int_{\rr^n} \Vert x\Vert ^2 \mu(dx)$, improving a well-known two-sided
estimate due to Bobkov. Our Markovian approach is remarkably simple and is
sufficiently robust to be extended beyond the log-concave case, at the price of
potentially modifying the underlying dynamics in the energy, leading to
weighted Poincar\'e inequalities. All our results are illustrated by some
classical and less classical examples.
