Abstract
We study the continuity, smoothing, and convergence properties of Steiner symmetrization in higher space dimensions. Our main result is that Steiner symmetrization is continuous in W 1, p $ (1 \leq p < \infty) $ in all dimensions. This implies that spherical symmetrization cannot be approximated in W 1, p by sequences of Steiner symmetrizations. We also give a quantitative version of the standard energy inequalities for spherical symmetrization.
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