Spectral gaps, symmetries and log-concave perturbations

Franck Barthe (1) , Boaz Klartag

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We discuss situations where perturbing a probability measure on R n does not deteriorate its Poincaré constant by much. A particular example is the symmetric exponential measure in 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 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.

Domaines: Analyse fonctionnelle [math.FA]; Probabilités [math.PR]

Franck Barthe, Boaz Klartag. Spectral gaps, symmetries and log-concave perturbations. 2019. ⟨hal-02170770⟩