In this paper, the extremal problem, min{ () : , GL()} p l K o K L n φ φ φ ∈ ⊆ ∈ , of two convex bodies K and L in n 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: 1 2 give the geometric distance between the unit ball 2 n B and a centrally symmetric convex body K.
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