A Set-Valued Extension Of The Mazur-Ulam Theorem

In this doctoral dissertation,we are devoted to studying the Mazur-Ulam theorem of isometric mappings from R(X)to R(Y)and C(X)to C(Y),where X and Y are Banach spaces,R(X)and C(X)denote the set of all non-empty compact convex and non-empty bounded closed convex sets of X,respectively,endowed with Hausdorff metrics.The famous Mazur-Ulam(1932)theorem states that " Every surjective isometry between Banach spaces must be affine",which profoundly reveals that the mapping of "metric-preserving" mappings must be "linearity-preserving".As a starting point,the research on "metric-preserving" or "isometric" mappings has been carried out for more than 80 years,and various theories have been promoted(such as those of non-surjective isometry,perturbed isometry or ε-isometry,coarse isometry,etc.)and their applications in related fields(for example,in the field of coarse geometry)has shown many fascinating and beautiful results.However,there are not many results on the properties of "set-valued" isometric mappings.Apart from a few results for some finite-dimensional spaces including its localizations,by the understanding of the author,the only one article on infinite dimensional spaces was by Zhou Yu et al.This is also the starting point of the motivation of this article.The reason is mainly due to the limitation of the "representation" tools of set-valued mappings.We use isometric embedding theorems for these special metric spaces developed in recent years,and the differentiability theory of convex functions to overcome these difficulties.The following main results of this paper are proved.Theorem A.Suppose X,Y are Banach spaces,and T:R(X)→R(Y)is a surjective isometry.Then T|X,the restriction of T to X must be an affine surjective isometry from X to Y.In particular,if one of the following two conditions is satisfied,i)One of X,Y is Gateaux smooth;ii)One of X,Y is strictly convex;then T(K)=U{Tx:x∈K},?K∈R(X).Theorem B.Suppose one of Banach space X,Y is an Asplund space,T:C(X)→C(Y)is a surjective isometry.Then the restriction T|X of T to X must be an affine surjective isometry from X to Y.In particular,if one of the following two conditions is satisfied,ⅰ)One of X,Y is Frechet smooth;ⅱ)One of X,Y is a locally uniform convex Asplund space;then T(C)=U{Tx:x∈C},? C∈C(X).Therefore,these results have affirmatively answered the two questions raised by Zhou et al.but also are new even if they are placed in finite-dimensional spaces.