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Inequalities for distances between points and distance preserving mappings. (English) Zbl 1076.51007

The authors consider a real (or complex) inner product space \(H\); they consider also the norm generated by the inner product. The parallelogram law states that the equality \[ \| y-x \|^2 +\| z-y \|^2 + \| w-z \|^2 + \| x-w \|^2 = \| z-x \|^2 + \| w-y \|^2 \] holds true if and only if \(y-x\), \(z-y\), \(w-z\), \(x-w\) are the sides of a (possibly degenerate) parallelogram with diagonals \(z-x\) and \(w-y\). In this paper the author presents a generalization of the short diagonals lemma by proving a new inequality for distances between six points.
In 1970 Aleksandrov had raised a question of whether a map \(f:X\longrightarrow Y\) between normed spaces preserving a distance \(\rho >0\) is an isometry. Beckman and Quarles knew in 1953 the answer to this question in the case of the \(n\)-dimensional Euclidean space when \(2\leq n<\infty\) and provided examples for non-isometric mappings which preserve unit distance for one-dimensional or for infinite dimensional real Euclidean spaces. In 1990 Rassias posed the following question: What happens if two (or more) distances are preserved by a map between normed spaces? Such a problem is called the Aleksandrov-Rassias problem.
In this paper the author also uses the generalization he obtained of the short diagonals lemma to investigate the Aleksandrov-Rassias problem in the particular cases where some distances are contractive and other distances are extensive.

MSC:

51K05 General theory of distance geometry
51M05 Euclidean geometries (general) and generalizations
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References:

[1] Aleksandrov, A. D., Mapping of families of sets, Sov. Math. Dokl., 11, 116-120 (1970) · Zbl 0213.48903
[2] Beckman, F. S.; Quarles, D. A., On isometries of Euclidean spaces, Proc. Amer. Math. Soc., 4, 810-815 (1953) · Zbl 0052.18204
[3] Benz, W., Isometrien in normierten Räumen, Aequationes Math., 29, 204-209 (1985) · Zbl 0588.46015
[4] Benz, W.; Berens, H., A contribution to a theorem of Ulam and Mazur, Aequationes Math., 34, 61-63 (1987) · Zbl 0651.46022
[5] Matoušek, J., Lectures on Discrete Geometry, (Graduate Texts in Mathematics, vol. 212 (2002), Springer: Springer New York) · Zbl 0999.52006
[6] Rassias, Th. M., Is a distance one preserving mapping between metric spaces always an isometry?, Amer. Math. Monthly, 90, 200 (1983) · Zbl 0512.54017
[7] Rassias, Th. M., Mappings that preserve unit distance, Indian J. Math., 32, 275-278 (1990) · Zbl 0737.51013
[8] Rassias, Th. M., Properties of isometries and approximate isometries, (Milovanovic, G. V., Recent Progress in Inequalities (1998), Kluwer: Kluwer Dodrecht), 341-379 · Zbl 0902.51013
[9] Xiang, S., Mappings of conservative distances and the Mazur-Ulam theorem, J. Math. Anal. Appl., 254, 262-274 (2001) · Zbl 0971.47001
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