TY - JOUR
T1 - Rate of convergence of geometric symmetrizations
AU - Klartag, B.
N1 - Funding Information:
Supported in part by the Israel Science Foundation and by the Minkowski Center for Geometry.
PY - 2004/12
Y1 - 2004/12
N2 - It is a classical fact, that given an arbitrary convex body K ⊃ ℝn there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we provide quantitative estimates regarding this convergence, for both Minkowski and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean ball, improving previously known exponential estimates. Inspired by a method of Diaconis [D], our technique involves spherical harmonics. We also make use of an earlier result by the author regarding "isomorphic Minkowski symmetrization".
AB - It is a classical fact, that given an arbitrary convex body K ⊃ ℝn there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we provide quantitative estimates regarding this convergence, for both Minkowski and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean ball, improving previously known exponential estimates. Inspired by a method of Diaconis [D], our technique involves spherical harmonics. We also make use of an earlier result by the author regarding "isomorphic Minkowski symmetrization".
UR - http://www.scopus.com/inward/record.url?scp=16244384177&partnerID=8YFLogxK
U2 - 10.1007/s00039-004-0493-4
DO - 10.1007/s00039-004-0493-4
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AN - SCOPUS:16244384177
SN - 1016-443X
VL - 14
SP - 1322
EP - 1338
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 6
ER -