5n Minkowski symmetrizations suffice to arrive at an approximate Euclidean ball

This paper proves that for every convex body in ℝⁿ there exist 5n Minkowski symmetrizations which transform the body into an approximate Euclidean ball. This result complements the sharp cn log n upper estimate by J. Bourgain, J. Lindenstrauss and V.D. Milman, of the number of random Minkowski symmetrizations sufficient for approaching an approximate Euclidean ball.