Extremal problems and isotropic positions of convex bodies

Abstract

LetK be a convex body in ℝn and letWi(K),i=1, …,n−1 be its quermassintegrals. We study minimization problems of the form min{Wi(TK)|T ∈ SLn} and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropic characterization for appropriate measures. This shows that several well known positions of convex bodies which play an important role in the local theory may be described in terms of classical convexity as isotropic ones. We provide new applications of this point of view for the minimal mean width position.