The minimal Orlicz mean width of convex bodies

In this paper, we generalize the minimal p-mean width of convex bodies (including the classic minimal mean width) to the Orlicz Brunn–Minkowski–Firey theory. The concept of Orlicz mean width of a convex body K in Rn, wϕ(K), is introduced. Then we study the minimization problems of the form min{wϕ(TK):T∈SL(n)} and show that bodies which appear as solutions of such problems satisfy isotropic conditions of a suitable measure. Finally, the characteristics of the condition for the minimum Mϕ(K)Mϕ∗(K) are obtained, a stability result for Lp-centroid bodies is established, and some new applications for the minimal Orlicz mean width position are provided.