This is an announcement for the paper "Convergence of random polarziations" by Almut Burchard and Marc Fortier.
Abstract: We derive conditions under which random sequences of polarizations converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose distributions may be far from uniform. The proof of convergence hinges on an estimate for the expected distance from the limit that also yields a bound on the rate of convergence. In the special case of i.i.d. sequences, we obtain almost sure convergence even for polarizations chosen at random from small sets. The precise characterization of convergent sequences remains an open problem. These statements about polarization allow us to improve the existing convergence results for Steiner symmetrization. In particular, we show that full rotational symmetry can be achieved by alternating Steiner symmetrization along directions that satisfy an explicit non-degeneracy condition. Finally, we construct examples for dense sequences of directions such that the corresponding Steiner symmetrizations do not converge.
Archive classification: math.FA math.PR
Mathematics Subject Classification: 60D05 (26D15, 28A75, 52A52)
Remarks: 30 pages, 6 figures
Submitted from: almut@math.toronto.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1104.4103
or