  We discuss situations where perturbing a probability measure on
$\mathbb{R}^n$ does not deteriorate its Poincar\'e constant by much. A
particular example is the symmetric exponential measure in $\mathbb{R}^n$, even
log-concave perturbations of which have Poincar\'e constants that grow at most
logarithmically with the dimension. This leads to estimates for the Poincar\'e
constants of $(n/2)$-dimensional sections of the unit ball of $\ell_p^n$ for $1
\leq p \leq 2$, which are optimal up to logarithmic factors. We also consider
symmetry properties of the eigenspace of the Laplace-type operator associated
with a log-concave measure. Under symmetry assumptions we show that the
dimension of this space is exactly $n$, and we exhibit a certain interlacing
between the "odd" and "even" parts of the spectrum.
