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We establish some new quantitative results on Steiner/Schwarz-type symmetrizations, continuing the line of related results of Bourgain, J. Lindenstrauss, J. and Milman, V. on Steiner symmetrizations (see, Bourgain, J. Lindenstrauss, J. and Milman, V., Estimates related to Steiner symmetrizations, GAFA 87-88 (Lindenstrauss, J. and Milman, V., eds.), vol. 1376, Springer Lecture Notes, 1989 pp. 264-273). We show that if we symmetrize high dimensional sections of convex bodies, then very few steps are required to bring such a body close to a Euclidean ball. We continue the study of high dimensional phenomena by introducing the limiting convolution body of two convex bodies extending the notion of the limiting convolution square studied by Kiener, K. and Schmuckenschlager, M. (see, Kiener, K., Extremalitat von Ellipsoiden und die Faltungsungleichung von Sobolev, Arch. Math. 46 (1986), 162-168 and Schmuckenschlager, M., The distribution function of the convolution square of a convex symmetric body in usd\IR\sp{n}usd, Israel Journal of Mathematics 78 (1992), 309-334). We compute examples demonstrating the diversity of bodies one may receive as a limiting convolution body which are of independent interest as well. However, we prove that for "generic" bodies the limiting convolution body is an ellipsoid.
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