It is well known that there exists a unique ellipsoid of maximal volume inside a convex body (a compact convex set with non-empty interiors) in R^n. This ellipsoid, now called the John ellipsoid named after Fritz John in 1948, has many applications in convex geometry, functional analysis, PDEs, optimization, and other subjects. There is a characterization of the John ellipsoid which plays a major role in the study of reverse isoperimetric inequalities.
    The goal of this talk is to describe the generalizations and applications of the classical John ellipsoid, including our recent work on generalizing the Lp ellipsoids to Orlicz setting and establishing the Orlicz version of Ball’s volume ratio inequality.