  This paper proves that for every convex body in R^n there exist 5n-4
Minkowski symmetrizations, which transform the body into an approximate
Euclidean ball. This result complements the sharp c n 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.
