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We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are truefor all log-concave measures that are rotationally invariant, extendingprevious results known for Gaussian measures. Actually, our result apply beyondthe case of log-concave measures, for instance to Cauchy measures as well. Forthe proof, new sharp weighted Poincar\'e inequalities are obtained for evenprobability measures that are log-concave with respect to a rotationallyinvariant measure.

Improved log-concavity for rotationally invariant measures of symmetric convex sets