Characterizations on Mazur–Ulam theorem

We consider the area preserving 22-isometries in linear 22-normed spaces. In Chu (2007) [10], the Mazur–Ulam theorem was proved in linear 22-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 22-normed spaces. Recently Amyari, Sadeghi [16], in non-Archimedean 22-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 22-normed spaces over a linear ordered non-Archimedean field without the strict convexity assumption.