球面空间凸性理论

本项目研究内容是近来分析学热点问题之一。凸性理论在线性空间中已获巨大成功，但受制于基本运算（变换）的匮乏和几何方法的局限性，在球面空间中的发展却十分缓慢；现今，大量实际应用或纯理论问题的解决又亟待系统球面凸性理论的建立与完善。因此，我们将采用以分析为主的方法，尝试建立系统的球面凸（几何）分析基础理论，并由此进行球面凸几何与球面凸函数方面的深入研究。具体开展球面凸集的组合、结构与分离性质；球面凸集类上基本运算与变换群；球面凸集几何不变量与赋值；球面凸函数基本性质及其应用等四部分的研究。目标是建立系统的球面凸集（函数）基础理论，研究解决球凸体实赋值的“Hadwiger型表示猜想”、“球凸体内蕴体积Urysohn、Blaschke-Santalo型不等式”、“球凸集浮动面积测度Minkowski型问题”等公开问题和球面凸优化等应用方面的问题。项目的完成将为本领域研究提供理论基础与有力工具。

This project concerns one of the popular topics in analysis. Convexity theory in linear spaces has gained great success, however, it develops very slowly in spherical spaces due to the lack of basic compositions (transformations) and the geometric method's limitation. Nowadays, the solutions of numerous practial or theoretical problems are waiting for the establishment of systematic convexity theory in spherical spaces. Therefore, we are going to establish a systematic basic theory of convex analysis and convex geometrical analysis in spherical spaces through an analytic approach mainly, and further to do deep research on spherically convex geometry and spherically convex functions. More precisely, for spherically convex sets, the following topics will be concerned: combinatorial, structural and seperation properties; basic compositions and transformations; geometric invariants and valuations; spherically convex function and its applications. The goal is the establishment of systematic basic theories of spherically convex sets and functions, the solutions of some open problems such as the conjeture of Hadwiger type for real-valued valuation on spherically convex sets, the inequalities of Urysohn type and Blaschke-Santalo ́ type for spherical intrinsic volumes and the problem of Minkowski type for spherical floating area measures etc, and also the applications of spherically convex functions, e.g., in spherically convex optimizations. The outcomes of this project will provide theoretical supports and powerful practical tools for further studies in this reasearch area.