  We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true
for all log-concave measures that are rotationally invariant, extending
previous results known for Gaussian measures. Actually, our result apply beyond
the case of log-concave measures, for instance to Cauchy measures as well. For
the proof, new sharp weighted Poincar\'e inequalities are obtained for even
probability measures that are log-concave with respect to a rotationally
invariant measure.
