On the Aleksandrov problem in linear -normed spaces

Chu, Hahng‐Yun; Lee, Keonhee; Park, Choonkil

doi:10.1016/j.na.2004.07.046

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Cited by 18 publications

(20 citation statements)

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“…In 2004, Chu et al [6] defined the concept of n-isometry which is suitable for representing the notion of n-distance preserving mappings in linear n-normed spaces and studied the Aleksandrov problem in linear n-normed spaces. For related works we refer the reader to [3,5,7] and [9].…”

Section: Introductionmentioning

confidence: 99%

Mappings of conservative distances in linear n -normed spaces

Chu

Choi

Kang

2009

Nonlinear Analysis: Theory, Methods & Applications

Self Cite

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Section: Introductionmentioning

confidence: 99%

Mappings of conservative distances in linear n -normed spaces

Chu

Choi

Kang

2009

Nonlinear Analysis: Theory, Methods & Applications

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“…It follows from (6), (7) and (8) This yields |α| = |β|. Moreover, |α| is a fixed positive real number.…”

Section: Proofmentioning

confidence: 86%

“…[4][5][6][7]). Chu et al [7] first generalized the Aleksandrov problem to n-normed spaces. Their main result [7,Theorem 2.10] proves that the weak n-distance one preserving mapping is an n-isometries under additional conditions (e.g.…”

Section: Introductionmentioning

confidence: 99%

Mappings of preserving n-distance one in n-normed spaces

Huang

Tan

2018

Aequat. Math.

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We give a positive answer to the Aleksandrov problem in n-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving n-distance one is affine, and thus is an n-isometry. This is the first time to solve the Aleksandrov problem in n-normed spaces with only surjective assumption even in the usual case n = 2. Finally, when the target space is n-strictly convex, we prove that every mapping preserving two n-distances with an integer ratio is an affine n-isometry.

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“…Since 2004, the Aleksandrov problem has been discussed in the n-normed spaces (n ≥ 2) (see [2,3,4,5]). Chu, Lee and Park proved:…”

Section: Introductionmentioning

confidence: 99%