Journal of the
Korean Mathematical Society JKMS

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 $f : H_{\theta} \to Y$, then the restriction $f|_{H_{\theta+\rho/2}}$ 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.

Keywords: Aleksandrov problem, isometry, distance-preserving mapping

MSC numbers: Primary 51K05