Extremal problems related to Gauss-John position

Abstract

In this paper, we consider the extremal problem of the lp-norm: min{lp(TK), ℴ∈TK⊆L,T∈GL(n)}, where K,L are two convex bodies in ℝn. Using the optimization theorem of John, we give necessary conditions for K to be in extremal position in terms of a decomposition of the identity. Furthermore, the weaker optimization problem, min{(lp(TK))p: TK ⊆ B2n, TK∩Sn−1 ≠ ∅, T∈GL(n)}, is also considered. As an application, the geometric distance between the unit ball B2n and a centrally symmetric convex body K is obtained.