Spectral gaps, symmetries and log-concave perturbations

We discuss situations where perturbing a probability measure on R n does not deteriorate its Poincare constant by much. A particular example is the symmetric exponential measure in R n , even log-concave perturbations of which have Poincare constants that grow at most logarithmically with the dimension. This leads to estimates for the Poincare constants of (n/2)-dimensional sections of the unit ball of l n p for 1 ≤ p ≤ 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.