Abstract.
Let X be a real Hilbert space with dim X ≥ 2 and let Y be a real normed space which is strictly convex. In this paper, we generalize a theorem of Benz by proving that if a mapping f, from an open convex subset of X into Y, has a contractive distance ρ and an extensive one Nρ (where N ≥ 2 is a fixed integer), then f is an isometry.
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Jung, SM. A characterization of isometries on an open convex set. Bull Braz Math Soc, New Series 37, 351–359 (2006). https://doi.org/10.1007/s00574-006-0015-0
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DOI: https://doi.org/10.1007/s00574-006-0015-0