凸浮体的相关问题

The floating bodies originate from the ancient Greek mathematician Archimedes. Recently, the convex form is reconsidered as a closely related area to the affine surface area by Schutt and Werner. We will consider the following problems in this research: firstly, to study the equilibrium problem of the floating bodies by geometric tomography; secondly, to study the uniqueness of convex bodies from the information of their floating bodies by spherical harmonics; finally, to study the "dual" problem between the floating bodies and the illuminating bodies. These results are important for the development to the convex floating bodies.

浮体问题最早起源于古希腊数学家阿基米德，然而近期Schutt教授和Werner教授将其凸体相关问题与仿射表面积问题紧密联系在一起。本项目拟运用凸几何分析里面的断层学理论主攻以下问题：首先利用几何断层学理论研究浮体的平衡理论；其次应用球面调和方法研究凸体是否被其浮体唯一确定问题；最后研究浮体与照明体的“对偶”问题。这些结论对发展和完善凸几何分析中的浮体问题起到至关重要的作用，在理论和应用上具有重大意义。

浮体起源于阿基米德，近期被Ulam、Gardner、Schneider、Schutt、Werner等人进一步研究，提出仿射表面积、浮体唯一性、浮体稳定性等一系列问题。本项目发现非中心凸体无法被自身凸浮体唯一确定；解决了对称凸体均衡问题；提出了Ulam浮函数的相关概念和性质；部分解决了log凹函数的仿射等周不等式问题；利用浮体逼近解决了可求长链的泛簇弱收敛问题。这些结论对浮体的进一步研究与应用提供了很好的理论支持。