In this paper, the extremal problem, min, of two convex bodies K and L in ℝⁿ is considered. For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the optimization theorem of John. Besides, we also consider the weaker optimization problem: min. As an application, we give the geometric distance between the unit ball B₂ⁿ and a centrally symmetric convex body K.