Characterizations on Mazur-Ulam theorem

Abstract

We consider the area preserving 2-isometries in linear 2-normed spaces. In Chu (2007) [10], the Mazur-Ulam theorem was proved in linear 2-normed spaces with the assumption of the collinearity preserving. In this article, we prove the Mazur-Ulam theorem for the interior preserving mappings in linear 2-normed spaces. Recently Amyari, Sadeghi [16], in non-Archimedean 2-normed spaces, the Mazur-Ulam theorem was proved under the condition of strict convexity. In this paper, we also prove the theorem on non-Archimedean 2-normed spaces over a linear ordered non-Archimedean field without the strict convexity assumption. (C) 2009 Elsevier Ltd. All rights reserved.