We shall discuss the conjectured logarithmic Brunn-Minkowski inequality of Böröczky, Lutwak, Yang and Zhang (2012), which is a far-reaching refinement of the classical Brunn-Minkowski inequality for symmetric convex sets. After a quick recap of known special cases, we will explain an equivalent local form of the conjecture which is a functional inequality for functions defined on the boundary of symmetric convex sets. Time permitting, we will then show the solution (from joint work with G. Moschidis) of the Gardner-Zvavitch problem, which is formally weaker than the log-Brunn-Minkowski inequality.
