运动公式与向量赋值问题研究

The kinematic formulas in integral geometry describe the relationship among the invariants between the intersected submanifold of two compact manifolds and the two manifolds. Some important geometry inequalities in integral geometry and convex geometric analysis, such as Minkowski inequality, Aleksandrov-Fenchel inequality, can be obtained by using the kinematic formulas. By means of kinematic density in homogeneous space, moving frame approach and the theory of valuation, we will give a description of kinematic formulas for the total mean curvature ellipsoid of submanifold, and we investigate a completely new kinematic formula - the kinematic formula for matrix. This is gerneralization from kinematic formulas for scalar invariants to those for matrix invariants as well as an extension of the scope of kinematic formulas. This is a new direction in the study of kinematic formulas. On the other hand, although mathematicians have established a characterization of SL(n) invariant valuation on polytopes, one is still wondering if there exits characterizations on vector valuations and tensor valuations. However there are few research results with respect to vector valuations. In this project, the classification of SL(n)-covariant valuations on the spaces of polytopes will be investigated by using techniques s the valuation extension, the dissection of polytopes, the solution to Cauchy equation and the theory of transformation group. This can be regarded as the first step towards a.complete classification of SL(n)-covariant tensor valuations.

积分几何中的运动公式刻画了空间中两紧子流形的相交子流形与两流形的不变量之间的关系。运用运动公式可得到积分几何与凸几何分析中包括Minkowski不等式、Aleksandrov-Fenchel不等式等一系列非常重要的几何不等式。本课题拟运用齐性空间运动密度、活动标架法、赋值理论等研究子流形的全平均曲率矩阵的运动公式，探讨矩阵型运动公式这一新的运动公式。这是传统的纯量型运动公式的推广，也是运动公式理论发展方向之一。另一方面，虽然前人已经给出了多胞形上的SL(n)不变的实值赋值的完全分类，但人们想进一步考虑其在更广泛的向量赋值、张量赋值的分类结果。目前相关的向量赋值的结果甚少。鉴于此，本项目拟运用赋值延拓、多胞形分割、取凸包、Cauchy泛函方程的解等赋值方法和变换群理论，探索多胞形上的SL(n)协变的向量赋值的分类刻画。这可以看作研究SL(n)协变的张量赋值的第一步。

应用运动公式可以得到积分几何与凸几何分析中包括Minkowski不等式、等周不等式等一系列非常重要的几何不等式。我们主要通过平移的Blaschke运动公式和Poincare公式，获得了Bonnesen型对称混合位似不等式，其中Bonnesen型对称混合位似不等式是经典等周不等式的推广；进一步地，通过包含测度和Blaschke滚动定理，我们得到逆Bonnesen型对称混合位似不等式。逆形式的Bonnesen型对称混合位似不等式是著名Bottema不等式的类似替代。同时，我们也研究具有几何背景的几何不等式与分析不等式，具体为研究平面上多面体的离散等周不等式的分析与几何形式。通过建立关于Schur凸和Schur凹函数的微分方程获得一系列离散的Bonnesen型不等式及逆Bonnesen型不等式。赋值理论起源于Dehn的Hilbert第三个问题的解，在积分几何与凸几何分析中扮演着重要角色。Ludwig、Reitzner等给出了多胞形上的SL(n)上不变的实值赋值的完全分类，受其主要思想启发，我们运用多胞形分割、Cauchy泛函方程的解、赋值延拓等赋值方法刻画多胞形上的SL(n)上不变的向量赋值的分类。获得了无任何连续性条件下的SL(n)协变的向量赋值完全分类。.