  Log-Brunn-Minkowski inequality was conjectured by Bor\"oczky, Lutwak, Yang
and Zhang \cite{BLYZ}, and it states that a certain strengthening of the
classical Brunn-Minkowski inequality is admissible in the case of symmetric
convex sets. It was recently shown by Nayar, Zvavitch, the second and the third
authors \cite{LMNZ}, that Log-Brunn-Minkowski inequality implies a certain
dimensional Brunn-Minkowski inequality for log-concave measures, which in the
case of Gaussian measure was conjectured by Gardner and Zvavitch \cite{GZ}.
  In this note, we obtain stability results for both Log-Brunn-Minkowski and
dimensional Brunn-Minkowski inequalities for rotation invariant log-conave
measures near a ball. Remarkably, the assumption of symmetry is only necessary
for Log-Brunn-Minkowski stability, which emphasizes an important difference
between the two conjectured inequalities.
  Also, we determine the infinitesimal version of the log-Brunn-Minkowski
inequality. As a consequence, we obtain a strong Poincar\'{e}-type inequality
in the case of unconditional convex sets, as well as for symmetric convex sets
on the plane.
  Additionally, we derive an infinitesimal equivalent version of the
B-conjecture for an arbitrary measure.
