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 The Aleksandrov problem and the Mazur-Ulam theorem on linear $n$-normed spaces Bull. Korean Math. Soc. 2013 Vol. 50, No. 5, 1631-1637 https://doi.org/10.4134/BKMS.2013.50.5.1631Published online September 1, 2013 Ma Yumei Dalian Nationality University Abstract : This paper generalizes the Aleksandrov problem and Mazur Ulam theorem to the case of $n$-normed spaces. For real $n$-normed spaces $X$ and $Y$, we will prove that $f$ is an affine isometry when the mapping satisfies the weaker assumptions that preserves unit distance, $n$-colinear and 2-colinear on same-order. Keywords : $n$-DOPP, $n$-isometry, $n$-Lipschitz, $2$-collinear, $n$-collinear MSC numbers : 46B04, 46B20, 51K05 Downloads: Full-text PDF