An inequality for distances between \(2n\) points and the Aleksandrov--Rassias problem
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.
scientific article; zbMATH DE number 5073029
scientific article