  In a recent work, Klartag gave an improved version of Lichnerowicz' spectral
gap bound for uniformly log-concave measures, which improves on the classical
estimate by taking into account the covariance matrix. We analyze the equality
cases in Klartag's bound, showing that it can be further improved whenever the
measure has no Gaussian factor. Additionally, we give a quantitative
improvement for log-concave measures with finite Fisher information.
