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

Abstract

For origin-symmetric convex bodies in ${\mathbb{R}}^n$ invariant under $n-2$ orthogonal hyperplanes reflections, we establish the logarithmic Brunn--Minkowski inequality and classify all equality cases without regularity assumptions. We derive the logarithmic Minkowski inequality and rigidity for cone-volume measures, including an explicit classification for bodies of revolution and uniqueness for simplicial polytopes in the prescribed symmetry class.