  We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries
to $n$ independent hyperplanes, and discuss the equality case and the
uniqueness of the solution of the related case of the logarithmic Minkowski
problem. We also clarify a small gap in the known argument classifying the
equality case of the log-Brunn-Minkowski conjecture for unconditional convex
bodies.
