Banach空间单位球面上的等距延拓及相关问题

The properties of the unit spheres of Banach spaces have important significance for the research of Banach space theory and its geometrical strucuture. The project applicant has been studying the isometric extension problem between Banach spaces for nearly twenty years. His research work gave many detailed characterizations of the isometries between unit spheres of classical Banach spaces and obtained many good methods and results as well. Last year, Professor E.Odell who is a well-known scholar of Banach space theory pointed out that isometric extesnion problem is "a very difficult problem that remains unsolved after 25 years"(see appendix 1). By now the project applicant and his students have solved this problem between any classical Banach space and a Banach space. Since a general Banach space does not have a specific form of the norm, some previous methods and techniques are not useful any more. Therefore, this project will mainly use the gemometrical properties of Banach spaces and related numerical index character to continue the study, and strive for a large progress of this problem in the cases of polyhedral Banach spaces, finite-dimensional Banach spaces, very "flat"(smooth) spaces and uniformly convex spaces. Moreover,this project will also use the previous experience to study the extension problem of Lipschitz mappings and numerical index theory in Banach spaces, and strive to make a big progress as well.

Banach空间单位球面性质对于研究空间理论及其几何结构有着重要的意义。项目申请人近二十年来一直坚持研究球面间的"等距延拓"问题，对很多经典Banach空间球面间的等距映射做了细致的刻画与研究，开创了一系列解决相关问题的方法，并获得许多好结果。去年，空间理论权威教授E.Odell教授曽说："等距延拓问题是一个25年尚未解决的非常困难的问题"（见附件1）。至今为止，申请人及其指导的学生们已经将所有"经典Banach空间到任意空间"的该问题完全解决。而对于一般的情况，由于没有范数的表达形式，以前的方法和技巧就失去了效用。因此，本项目将试图利用球面的一些几何性质及有关的数值指标特征来继续研究，争取在多面体空间，有限维空间，足够"平坦"（光滑）空间以及一致凸空间等一般情况下取得较大突破。同时，本项目也将进一步研究Lipschitz映射延拓及空间数值指标理论等问题，力争在这些相关领域均取得较大进展。

Banach空间单位球面性质对于研究空间理论及其几何结构有着重要的意义。项目负责人近二十年来一直坚持研究球面间的"等距延拓"问题，对很多经典Banach空间球面间的等距映射做了细致的刻画与研究。本项目试图研究一般赋范空间的"等距延拓"问题，由于没有范数的表达形式，以前的方法和技巧就失去了效用。本项目通过研究赋范空间上球面的几何性质去解决"等距延拓"问题，并在二维空间、严格凸空间和lush空间等取得突破性进展。此外，借鉴这些研究成果和研究方法，我们在与"等距延拓"问题密切相关的Aleksandrov问题和Wigner定理取得一些好的研究成果。同时，这些研究成果也被推广到n-赋范空间，此空间上n-范数是面积和体积的推广。对球面几何性质的研究，使得我们在空间数值指标和Lipschitz映射延拓等方面取得较大进展。本项目共完成论文25篇，已经发表的19篇论文中有14篇SCI论文，接受待发表的3篇论文, 还有3篇处于投稿状态。