Near-Isometries of the Unit Sphere
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BRIEF COMMUNICATIONS NEAR-ISOMETRIES OF THE UNIT SPHERE I. A. Vestfrid
UDC 517.5
n We approximate "-isometries of the unit sphere in `n 2 and `1 by linear isometries.
1. Introduction Notation. Throughout the paper, X and Y denote real normed spaces. A sphere and a closed ball with center z and radius r are denoted by S(z, r) and B(z, r), respectively. We also write S(0, r) = S(r) and B(0, r) = B(r). The unit sphere and the ball are denoted by S and B (or SE and BE when it is necessary to specify the space), respectively. For a point x in Rn , xi denotes its ith coordinate in the standard basis {ei }ni=1 . A local version of the classical Mazur–Ulam theorem asserts that a local isometry f, which maps an open connected subset of X onto an open subset of Y, is the restriction of an affine isometry of X onto Y (see, e.g., [1, p. 341]). This classical result was generalized in several directions. One of them is the study of the problem of isometric extension posed by Tingley [8]: Let T be a surjective isometry between the spheres of X and Y. Is T necessarily the restriction of a linear isometry between X and Y ? There are numerous publications devoted to Tingley’s problem (see [2] for a survey of the corresponding results) and, in particular, the problem is positively solved for various specific classical Banach spaces. If the distances are not known precisely, then it is natural to study how close f is to an isometry. There are various useful concepts of approximate isometry and, hence, one may ask whether a mapping of this kind, which preserves distances only approximately can be well approximated by a true isometry, especially by an affine isometry (see [1], Chapters 14 and 15, and the surveys [6] and [7] for a more complete presentation and the literature on this subject). Definition. Let A be a subset of X and let " ≥ 0. A map f : A ! Y is called an "-isometry if
for all x, y 2 A.
� � �kf (x) − f (y)k − kx − yk� "
(1)
The author [9, 11] presented sharp results on the approximation of "-isometries of balls in `n2 and `n1 . In the present paper, we study the approximations of "-isometries of spheres in `n2 and `n1 . We present the following results proceeding the way of [9, 11]: Israel Institute of Technology, Haifa, Israel; e-mail: [email protected]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 575–580, April, 2020. Original article submitted March 20, 2018. 0041-5995/20/7204–0663
© 2020
Springer Science+Business Media, LLC
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I. A. V ESTFRID
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Theorem 1. Let f : S`n2 ! S`n2 be an "-isometry. Then there is a linear isometry U of `n2 such that kf (x) − U xk C log(n + 1)",
x 2 S`n2 ,
(2)
for some absolute constant C. The upper bound in (2) is sharp. Theorem 2. There are absolute constants C and c with the following property: Let 0 < " < c and f : S`n1 ! S`n1 be an "-isometry. Then there is a unique linear isometry U of `n1 such that kf (x) − U xk C",
x 2 S`n1 .
(3)
2. Proofs We need the following lemma, which is, in fact, proved by the proof of Lemma 6 in [9]: L
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