Abstract
This paper proves that for every convex body in $\mathbb{R}^n$ 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.
