Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON DISTANCE-PRESERVING MAPPINGS

  • Jung, Soon-Mo (Mathematics Section College of Science and Technology Hong-Ik University) ;
  • M.Rassias, Themistocles (Department of Mathematics National Technocal University of Athens)
  • Published : 2004.07.01
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Abstract

We generalize a theorem of W. Benz by proving the following result: Let $H_{\theta}$ be a half space of a real Hilbert space with dimension $\geq$ 3 and let Y be a real normed space which is strictly convex. If a distance $\rho$ > 0 is contractive and another distance N$\rho$ (N $\geq$ 2) is extensive by a mapping f : $H_{\theta}$ \longrightarrow Y, then the restriction f│$_{\theta}$ $H_{+}$$\rho$/2// is an isometry, where $H_{\theta}$＋$\rho$/2/ is also a half space which is a proper subset of $H_{\theta}$. Applying the above result, we also generalize a classical theorem of Beckman and Quarles.

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References

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