Random polarizations

Details Random polarizations Random polarizations

2011-04-20

arxiv.org/abs/1104.4103v3

Mathematics

Functional Analysis

28 pages, 6 figures. Majorly re-organized for Version 3. Lemma 4.5 was false in the previous version (the error occurs in the

Scientific paper

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. These statements about polarization allow us to improve the existing convergence results for Steiner symmetrization. In particular we find bounds on the rate of convergence that require no convexity assumptions. We further show that full rotational symmetry can be achieved by alternating Steiner symmetrization along finitely many 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.

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