Minimal surface area position of a convex body is not always an M-position

Abstract

Milman proved that there exists an absolute constant C > 0 such that, for every convex body K in ℝn, there exists a linear image TK of K with volume 1, such that |TK + Dn|1/n ≤ C, where Dn is the Euclidean ball of volume 1. TK is then said to be in M-position. Giannopoulos and Milman asked if every convex body that has minimal surface area among all its affine images of volume 1 is also in M-position. We prove that the answer to this question is negative, even in the 1-unconditional case.