The Aleksandrov Problem And Mazur-Ulam Theorem On Quasi Convex N-Normed Linear Spaces

In this paper,we review the introduction and status of the Aleksandrov problem、Mazur-Ulam theorem and Aleksandrov-Rassias problem on normed linear spaces and m-normed linear spaces（m = 2,n）.Moreover,we introduce the definition of quasi convex n-normed linear space,and also research these questions above mentioned,get some conclusions on it.In the first section,we mainly review the introduction and results about Aleksandrov problem、Mazur-Ulam theorem on normed linear spaces and m-normed linear spaces.It is the conclusions on m-normed linear spaces in literatures[13]and[14]obtained by researcher H.Y.Chu that we have to pay particular attention to.In the seond section,we discuss the Aleksandrov problem and prove the Mazur-Ulam theorem on quasi convex n-normed linear spaces.We obtain f is an n-isometry only the conditions of f satisfies nDOPP and preserves 2-collinearity,also get the n-isometry mappings is affine mappings.In the third section,we mainly discuss the Aleksandrov-Rassias problem on quasi convex n-normed linear spaces.Basing on the literatures[33]and[44]which have got some conclusions about Aleksandrov-Rassias problem on normed linear spaces and n-normed linear spaces,we prove the conclusions are still success while the condition||x₁-y₁,x₂-y₂,…,x_n-y_n||≥1（?）||f（x₁）-f（y₁）,f（x₂）-f（y₂）,…,f（x_n）-f（y_n）||≥₁ replaced by||x₁-y₁,x₂-y₂,…,x_n-y_n||≤1（?）||f（x₁）-f（y₁）,f（x₂）-f（y₂）,…,f（x_n）-f（y_n）||≤₁.