Abstract
We establish some new quantitative results on Steiner/Schwarz-type symmetrizations, continuing the line of results from [Bourgain et al. (Lecture Notes in Math. 1376 (1988), 44–66)] on Steiner symmetrizations. 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Busemann, H.: Volume in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1–12.
Bourgain, J., Lindenstrauss, J. and Milman, V.: Minkowski sums and symmetrizations, in J. Lindenstrauss, and V. Milman, (eds), GAFA 86–87, Lecture Notes in Math. 1317, Springer-Verlag, 1988, pp. 44–66.
Bourgain, J., Lindenstrauss, J. and Milman, V.: Estimates related to Steiner symmetrizations, in J. Lindenstrauss, and V. Milman, (eds), GAFA 87–88, Lecture Notes in Maths 1376, Springer-Verlag, 1989, pp. 264–273.
Bourgain, J. and Milman, V.: New volume ratio properties for convex symmetric bodies in ℝn, Invent. Math. 88 (1987), 319–340.
Burago, Y. and Zalgaller, V.: Geometric Inequalities, Springer's Series in Soviet Math. 285 Springer-Verlag, 1988.
Figiel, T., Lindenstrauss, J. and Milman, V.: The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53–94.
Macbeath, A. M.: An extremal property of the hypersphere, Proc. Camb. Phil. Soc. (1951), 245–247.
Mani-Levitska, P.: Random Steiner symmetrizations, Studia Sci. Math. Hung. 21 (1986), 373–378.
Meyer, M. and Pajor, A.: On Santalo's inequality, in: J. Lindenstrauss and V. Milman, (eds), GAFA 87–88, Lecture Note in Maths 1376, Springer-Verlag, 1989, pp. 261–263.
Milman, V. and Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: J. Lindenstrauss, and V. Milman, (eds), GAFA 87–88, Lecture Notes in Maths 1376, Springer-Verlag, 1989, pp. 64–104.
Milman, V. and Schechtman, G.: Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Math. 1200, Springer-Verlag, 1986.
Pisier, G.: The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, 94 Cambridge Univ. Press, 1989.
Rogers, C. A. and Shephard, G. C.: The difference body of a convex body, Arch. Math. 8 (1957), 220–233.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tsolomitis, A. Quantitative Steiner/Schwarz-type symmetrizations. Geom Dedicata 60, 187–206 (1996). https://doi.org/10.1007/BF00160622
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00160622