The conjectured logarithmic Brunn-Minkowski inequality is a stronger version of the classical Brunn-Minkowski inequality, that has many applications in convex geometry, stochastis and other fields of mathematics. In my recent work, I prove a local version of the logarithmic Brunn-Minkowski inequality for one body of revolution and
one arbitrary body. Equality cases are discussed and some consequences such as the logarithmic Minkowski inequality and the uniqueness of the cone volume measure for bodies of revolution are deduced. The proof uses an operator theoretic approach together with the decomposition of spherical functions into isotypical components with respect to rotations around a fixed axis
