On non linear perturbations of isometries

“…A subsequent joint paper with S. Ulam [81] proposed the more delicate question of the stability of the equation defining isometries; more precisely: If f is a rough isometry, is it close to an isometry? After many partial results spanning half a century (starting with the Hilbertian case in [81]), the general case was solved affirmatively by P. Gruber [73] and J. Gevirtz [62] and the sharp constant provided by M. Omladič and P. Šemrl [122] in 1995.…”

Section: Quasificationmentioning

confidence: 99%

An invitation to bounded cohomology

Monod¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

View full text Add to dashboard Cite

show abstract

“…He showed that if, in Theorem 3.3 above, the space X is Euclidean, then the bounds 3ε and 5ε may be replaced by 3ε/ √ 2 and 5ε/ √ 2, respectively. Recently, Omladič and Šemrl [18] have shown that indeed these bounds can be reduced. Their main result is the following theorem.…”

Section: Case 1 (D > 18ε) Let the Real Number T Be Defined By D2mentioning

confidence: 99%

“…The study of ε-isometries of Banach spaces was revived by Bourgin [6] and Bourgin [5], Omladič and Šemrl [18]. Gruber [13] demonstrated the stability of surjective isometries between all finite dimensional normed vector spaces.…”

Section: Theorem 22 Let S 1 and S 2 Be Compact Metric Spaces If T mentioning

confidence: 99%

Isometries and approximate isometries

Rassias

2001

International Journal of Mathematics and Mathematical Sciences

View full text Add to dashboard Cite

Abstract. Some properties of isometric mappings as well as approximate isometries are studied.2000 Mathematics Subject Classification. Primary 46B04. Mazur and Ulam [17] proved the following well-known result concerning isometries, that is, transformations which preserve distances. Isometry and linearity. Theorem Given two real normed vector spaces X and Y , let U be a surjective mapping from X onto Y such that U(x)−U(y) = x −y for all x and y in X. Then the mapping x U(x)− U(0) is linear.Since continuity is implied by isometry, the proof of this theorem consists of showing that U(x) − U(0) is additive, and the additivity of this mapping will follow if we can prove that U satisfies Jensen's equation:In general this is not easy. However, for the special case in which the space Y is strictly convex, the proof is simple (see Baker [1]). A (real) normed space is called strictly convex if, for each pair of its nonzero elements y, z such that y +z = y + z , it follows that y = cz for some real number c > 0. When Y is strictly convex, it is easy to show that the unique solution to the two equations

show abstract

On non linear perturbations of isometries

Cited by 108 publications

References 12 publications

Stability of Surjectivity

Stability of Surjectivity

An invitation to bounded cohomology

Isometries and approximate isometries

Contact Info

Product

Resources

About