On Mazur-Ulam theorem and mappings which preserve distances (Q2709186)

Let \(f:X\to Y\) be a one-to-one mapping between normed vector spaces \(X\) and \(Y.\) The mapping \(f\) preserves the distance \(r>0\) if \(\|x-y\|=r\) implies that \(\|fx-fy\|=r\) .NEWLINENEWLINENEWLINEIf \(f\) preserves every distance \(r>0\) then \(f\) is linear up to translation according the well known theorem of \textit{S. Mazur} and \textit{S. Ulam} [C. R. Acad. Sci. Paris 194, 1115-1118 (1932; Zbl 0004.02103)]. It is an open question: whether or not \(f:\ell_{2}\to \ell_{2}\) must be linear if it preserves two distances (say, \(r_{1}\) and \(r_{2}\)) with a noninteger ratio \(r_{1}/r_{2}\) ? NEWLINENEWLINENEWLINEIn the present paper is proved that answer is ``yes'' in a special case \(r_{1}/r_{2}=n(4^{m}k^{2}-(4^{m}-1)/3)^{1/2}\) for some positive integers \(n, m\) and \(k\) .