It is known that there exists a unique ellipsoid of maximal volume inside a convex body in. This ellipsoid is called the John ellipsoid (named after mathematician Fritz John), and has many applications in convex geometry, functional analysis, and optimizations, etc. Particularly, that combining the isotropic characterization of John ellipsoid with the celebrated Brascamp-Lieb inequality in real analysis, plays a major role to attack reverse isoperimetric problem.

In this talk, we will first briefly survey the discipline of convex geometry. Then, the classical John ellipsoid and its applications to affine isoperimetric inequalities will be discussed. Finally, we will introduce our recent work on (reverse) affine isoperimetric problems.