An inequality for distances between \(2n\) points and the Aleksandrov--Rassias problem
type: Item
instance of: scholarly article
title: An inequality for distances between \(2n\) points and the Aleksandrov--Rassias problem
zbMATH Open document ID: 1109.46027
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: The authors first prove an inequality for the distance between two points among the given \(2n\) points of a Hilbert space \(H\). Then they consider the so-called Aleksandrov--Rassias problem, dealing with the notions for distances of being contractive, extensive, preserved by \(f : H_1 \rightarrow H_2\) in the case of real Hilbert spaces. We recall that a distance \(\rho\) is contractive, extensive or preserved by \(f\) if \(\| x-y\| = \rho\) implies \(\| f(x) - f(y)\| \leq (\geq) (=) \rho\). For an \(n\) smaller than or equal to the dimension of the Hilbert spaces, if the distances \(\{\alpha_{ij},\beta_{ij},\gamma_{ij}\}_{1\leq i,j \leq n}\) are contractive by \(f\) for \(i<j\) and \(\gamma_{ii}\) are extensive by \(f\), then \(f\) preserves all the distances for \(i\leq j\). As an interesting corollary, they deduce that for \(n \geq 3\), if a distance \(\rho\) is preserved, \(2^{-\frac{1}{2}}\rho\) is contractive and \(3^{\frac{1}{2}}\rho\) is extensive by \(f\), then \(f\) is a linear isometry up to translation. This last result uses Theorem 2.1 from \textit{S.~Xiang} [J.~Math.\ Anal.\ Appl.\ 254, 262--274 (2001; Zbl 0971.47001)], where a similar conclusion is obtained when \(\rho\) and \(3^{\frac{1}{2}} \rho\) are preserved.
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 46T99
zbMATH DE Number: 5073029
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: parallelogram law
zbMATH Keywords: distance preserving maps
author: Soon-Mo Jung
author: Ki-Suk Lee
reviewed by: T.S.S.R.K. Rao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.043
OpenAlex ID: W2093234270
cites work: Q5615895
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DOI: 10.1016/J.JMAA.2006.01.043
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description: scientific article; zbMATH DE number 5073029