Extension Of Isometries And Distance One Preserving Mappings

We recall that a mapping T from a metric space (E,dE) to another metric space (F,dE) is called 1-Lipschitz if dF(T(x),T(y))≤dE(x,y) for allx,y∈E. (0.2) T is called anti-1-Lipschitz if "≤" is replaced by "≥", and it is called an isometry if the equality holds in (0.2) for all x,y∈E.The classical Mazur-Ulam theorem [84] stated that any surjective isometry T be-tween two real normed spaces with T(0)=0 must be linear. Mankiewicz [83] ex-tended this result by showing that every isometry between open connected subsets of two normed spaces E and F can be extended to an affine isometry from E onto F. The two results show that E and F are linearly isometric if and only if their unit balls are isometric, which means that the linear structure of a normed space is completely de-termined by its unit ball as a metric space. A very natural set which one feels may determines the space is the unit sphere. In 1987, D.Tingley [117] posed the following problem:Problem 0.1. Let E and F be two normed spaces. If T0 is a surjective isometry between the unit spheres S(E) and S(F), does T0 has an isometric affine extension on the whole space?It is called the isometric extension problem or also called Tingley problem. To this problem, we always consider the real case, because the answer is obviously negative in the complex case. For example, E=F=(?) and T0(x)= x for all x€(?) with|x|=1. Since there is no linear or even metrically convex structure on the unit sphere, it is hard to answer this problem. So far, it is still open in the general case. In [117] Tingley only got the affirmative answer on assertion T(-x)=-T(x) for each x∈S(E), when the spaces are of finite dimension. During the past decades, Professor G.Ding and his students have been kept on this topic and had obtained many significant results(see [27, 34,36,40] and references therein). So far, Tingley problem for the surjective isometries between unit spheres of the same type classical Banach spaces has almost been solved. There is a close relationship between nonexpansive (i.e.1-Lipschitz) maps and isometries. A direct compact argument shows that every nonexpansive map from a compact metric space onto itself must be an isometry. In recent years, the 1-Lipschitz map extension problem has been extensively studied(see [31,66,42] and references therein). In [31] Ding got an affirmative answer for 1-Lipschitz mappings between the unit spheres of two Hilbert spaces. Liu [66] obtained that non-surjective 1-Lipschitz mapping between the unit spheres of two normed spaces, under some conditions, can be extended to be a real linear isometric mapping on the whole space. Fang [42] give an affirmative answer to Tingley’s problem for 1-Lipschitz mappings between the unit spheres of lp(T) type spaces.In Chapter 2, we discuss the 1-Lipschitz mappings between the unit spheres of vector valued spaces Lp(Ω,Σ,μ;Lq(X,(?),v)) (2≤q< p∞) and anti-1-Lipschitz mappings between the unit spheres of vector valued spaces Lp(Ω,Σ,μ;Lq(X,(?),v)) (1<p<q< 2) and obtain that, every such mapping can be extended to be a real linear isometry on the whole space LP(Ω,Σ,μ;Lq(X,(?), v)). Moreover, we study the extension problem of isometries between the the unit spheres of lp-sum (1< p<∞,p≠2) of lq type spaces, and give an affirmative answer to Tingley’s problem in this case.Let (E, dE) and (F, dF) be metric spaces. For some fixed number r> 0, we call that ? preserves distance r if for all x,y∈E with dE(x,y)=r, we have dF(?(x),?(y))= r and r is called a conservative distance for the mapping ?.In 1970, A.D.Alexandrov raised the well-known problemProblem 0.2. Whether or not a mapping with distance one preserving property is an isometry?Some results about this problem can be seen in [79-81,94-99].The concept of linear 2-normed space was introduced by S.Gahler [49], which is a generalization of the notion of normed space. H. Y.Chu [15] proved that the Mazur-Ulam theorem holds in real linear 2-normed spaces. In 1989, A.Misiak [86] introduced the concept of n-normed spaces which is a generalization of the concept of normed spaces and 2-normed spaces. H. Y.Chu [12] defined the notion of n-isometry which is suitable for representing the notion of n-distance preserving mappings in linear n-normed spaces.In Chapter 3, we provide some remarks on the Alexandrov problem in linear 2- normed spaces for the generalization of earlier results in [11]. In addition the author introduces the concept of linear (2,p)-normed spaces and for such spaces solves the corresponding Alexandrov problem. Furthermore we generalize the results to linear n-normed spaces as well as linear (n,p)-normed spaces.S.Mazur and S.Ulam [84] proved a well-known result that every isometry of a real normed vector space onto another real normed space is a linear mapping up to translation. The property is not true for complex normed vector spaces. The hypoth-esis of surjectivity is essential. Without this assumption Baker [5] proved that every isometry from a real normed space into a strictly convex normed space is linear up to translation. Moslehian [87] established a Mazur-Ulam type theorem in the frame-work of strictly convex non-Archimedean normed spaces. H.Y.Chu [12] proved that the Mazur-Ulam theorem holds in linear n-normed spaces. H.Y.Chu [13] proved that Riesz lemma hold in 2-normed spaces. In Chapter 4, we establish a Mazur-Ulam the-orem in non-Archimedean strictly convex 2-normed spaces and non-Archimedean p-strictly convex (2,p)-normed spaces, moreover generalize the results to strictly convex non-Archimedean n-normed spaces as well as p-strictly convex non-Archimedean (n,p)-normed spaces. We also study the existence and uniqueness of the best approximation in real linear 2-normed spaces and Riesz lemma in real linear n-normed spaces.