An isomorphic version of the slicing problem

Here we show that any centrally-symmetric convex body K⊂Rn has a perturbation T⊂Rn which is convex and centrally-symmetric, such that the isotropic constant of T is universally bounded. T is close to K in the sense that the Banach–Mazur distance between T and K is O(log n). If K is a body of a non-trivial type then the distance is universally bounded. The distance is also universally bounded if the perturbation T is allowed to be non-convex. Our technique involves the use of mixed volumes and Alexandrov–Fenchel inequalities. Some additional applications of this technique are presented here.