Steiner symmetrization $$(n-1)$$ ( n - 1 ) times is sufficient to transform an ellipsoid to a ball in $${\mathbb {R}}^
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Steiner symmetrization (n − 1) times is sufficient to transform an ellipsoid to a ball in Rn Yude Liu1 · Qiang Sun1 · Ge Xiong1 Received: 16 March 2020 / Accepted: 22 June 2020 © Fondation Carl-Herz and Springer Nature Switzerland AG 2020
Abstract In this article, we show that Steiner symmetrization (n − 1) times is sufficient to transform an ellipsoid to a ball in Rn . Specifically, we seek out the (n − 1) directions in the unit sphere such that the destination of the corresponding Steiner symmetrization is the standard ball. Keywords Steiner symmetrization · Ellipsoid · Ball Mathematics Subject Classification 52A20 Résumé Dans cet article, nous démontrons qu’il suffit de transformer un ellipsoïde en sphère dans R n en utilisant (n − 1) fois la symétrisation de Steiner. En particulier, on trouve les (n − 1) directions sur la surface de la sphère unitaire de telle sorte que la destination de la symétrisation de Steiner correspondante est une sphère standard.
1 Introduction The setting of this article is the n−dimensional Euclidean space, Rn . Let K be a compact convex subset of Rn . Given a unit vector u, view K as a family of line segments parallel to u. Slide these segments along u so that each is symmetrically balanced around the hyperplane u ⊥ . By the Cavalieri principle, the volume of K is unchanged by this rearrangement. The new set, called the Steiner symmetrization of K in the direction u, is denoted by Su K . Note that Su K is also convex, and Su K ⊆ Su L whenever K ⊆ L.
Research of the authors was supported by NSFC No. 11871373.
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Ge Xiong [email protected] Yude Liu [email protected] Qiang Sun [email protected]
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School of Mathematical Sciences, Tongji University, Shanghai 200092, People’s Republic of China
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Y. Liu et al.
The most important fact is that if one iterates Steiner symmetrizations of K through a suitable sequence of unit directions, the successive Steiner symmetrals of K will tend to a Euclidean ball in the Hausdorff topology on compact convex subsets of Rn . A detailed proof of this assertion can be found in [8, p. 98] or [12, p. 172]. It is this property that makes Steiner symmetrization a fundamental geometric method to attack isoperimetric problems and establish related affine isoperimetric inequalities for over 150 years. See, e.g., [1–7,9– 11,13–16,18] for references. In this article, we show that Steiner symmetrization (n − 1) times is sufficient to transform an ellipsoid to a ball in Rn . As highlighted below, we seek out the (n − 1) directions on the unit sphere Sn−1 of Rn such that the destination of the corresponding Steiner symmetrization is precisely the standard ball. Finally, we indicate practically how to transform an ellipsoid into a ball. Theorem 1.1 Let E be an ellipsoid in Rn . Then there exist u 1 , u 2 , . . . , u n−1 ∈ Sn−1 such that Su n−1 Su n−2 . . . Su 1 E is a ball.
2 Preliminaries Write B for the unit ball, i.e., the convex hull of the unit sphere Sn−1 in Rn . The volume of B is n
2 the constant ωn = (πn +1) . As usual, we use V to denot
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