Spectral gaps, symmetries and log-concave perturbations

Abstract

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

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