An isomorphic version of the slicing problem

Here we show that any n-dimensional centrally symmetric convex body K has an n-dimensional perturbation T 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 has a non-trivial type then the distance is universally bounded. In addition, if K is quasi-convex then there exists a quasi-convex T with a universally bounded isotropic constant and with a universally bounded distance to K.