平移协变的赋值

The theory on valuations goes back to Hilbert’s third problem at the beginning of the last century. The development of this theory relates closely to the convex geometric analysis. Its application can be found in many branches of Mathematics, such as integral geometry, functional analysis, differential geometry, algebraic geometry, complex geometry and information theory. Based on the dissection of basic geometric objects and the solution of the Cauchy type functional equations, this project is going to study Laplace transform on lattice polytopes, Ball operator on log-concave functions and the moment vector on convex bodies, to try to establish their characterizations as valuations and to further develope a general theory on the characterization of translation covariant valuations, which aims to contribute to the advance of valuation theory and convex geometric analysis.

赋值理论起源于二十世纪初的Hilbert第三问题，其发展与凸几何分析紧密相连，在积分几何、泛函分析、微分几何、代数几何、复几何和信息论等数学领域均有广泛的应用。本项目将从基本几何体的切割和Cauchy型函数方程的解入手，研究格点多胞形上的Laplace变换、对数凹函数上的Ball算子和凸体上的矩向量，力争给出它们的赋值刻画，并进一步发展平移协变赋值刻画的统一理论，旨在丰富赋值理论和推动凸几何分析理论的发展。

几何赋值理论起源于Hilbert第三问题，其100多年来的发展与凸几何分析紧密相连。本项目主要研究凸多胞形空间上的平移协变的赋值，得到了矩向量、矩矩阵和Lutwak-Yang-Zhang矩阵的SL(n)协（反）变赋值刻画，推动了几何赋值理论的发展，并为其在凸几何分析、积分几何、泛函分析、微分几何和信息论等相关学科领域中的应用提供了一定的理论和方法。