Logarithmic Brunn--Minkowski Inequality under $n-2$ Reflection Symmetries

We prove the log-Brunn-Minkowski inequality for origin-symmetric convex bodies with symmetries to n-2 orthogonal hyperplanes, and discuss the equality case and the uniqueness of the related logarithmic Minkowski problem. We also generalize the inequality from the Lebesgue measure to certain log-concave measure.