Abstract
It is known that given any convex bodyK ⊂ ℝn there is a sequence of suitable iterated Steiner symmetrizations ofK that converges, in the Hausdorff metric, to a ball of the same volume. Hadwiger and, more recently, Bourgain, Lindenstrauss and Milman have given estimates from above of the numberN of symmetrizations necessary to transformK into a body whose distance from the equivalent ball is less than an arbitrary positive constant.
In this paper we will exhibit some examples of convex bodies which are “hard to make spherical”. For instance, for any choice of positive integersn≥2 andm, we construct ann-dimensional convex body with the property that any sequence ofm symmetrizations does not decrease its distance from the ball. A consequence of these constructions are some lower bounds on the numberN.
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J. Bourgain, J. Lindenstrauss and V. Milman,Estimates related to Steiner symmetrizations, inGAFA 87-88 (J. Lindenstrauss and V. Milman, eds.), Lecture Notes in Mathematics1376, Springer, Berlin, 1989, pp. 264–273.
Yu. D. Burago and V. A. Zalgaller,Geometric Inequalities, Springer, Berlin, 1988.
G. Fejes Tòth and W. Kuperberg,Packing and coverings with convex sets, inHandbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.), North-Holland, Amsterdam, 1993, pp. 799–860.
H. Hadwiger,Einfache Herleitung der isoperimetrischen Ungleichung für abgeschlossen Punktmengen, Mathematische Annalen124 (1952), 158–160.
H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957.
H. Halberstam and K. F. Roth,Sequences, Oxford University Press, London, 1966.
A. Tsolomitis,Quantitative Steiner/Schwarz-type symmetrizations, Geometriae Dedicata60 (1996), 187–206.
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Bianchi, G., Gronchi, P. Steiner symmetrals and their distance from a ball. Isr. J. Math. 135, 181–192 (2003). https://doi.org/10.1007/BF02776056
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DOI: https://doi.org/10.1007/BF02776056