TITLE: Visual proof of convergence for Steiner's symmetrization
QUESTION [7 upvotes]: I want to find a visual proof of the following fact:

For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.

All proofs I know require some integral estimates.  I would prefer a more visual proof (even if it is more involved).

REPLY [3 votes]: Just for illustration, here is an animation of one step of a 
Steiner symmetrization
applied to a rasterized version of a polygon.

                    


          


          

First: a histogram of vertical lengths.
Second: lengths lifted to be symmetric around horizontal.