凸体的赋值理论是算子理论与凸几何学相结合的一个交叉学科。最近二十多年来，它是凸几何分析中特别活跃的热点领域。本项目主要研究: 1) Laplace变换的赋值刻画，尝试去发现可以用来刻画Laplace变换的最本质的性质；2) 凸体到连续函数空间的赋值的分类，这样的赋值应包含Lp质心体、Lp投影体以及Laplace变换；3) Haberl和Parapatits测度值赋值的对偶问题， 刻画对偶Lp表面积测度；4) 双赋值的Hadwiger定理，刻画混合体积。 在研究过程中，我们将把凸体的赋值理论、Fourier变换，局部渐进理论和调和分析理论等结合起来，并采用比较研究的方法，即深入分析各个领域数学家们所提出的研究方法，分析其产生的背景和使用的条件，力争综合运用这些方法并加以发展和创新。

The valuations theory in convex geometry is a subject combining operator theory and convex geometry. In recent two decades, the valuations theory becomes one of the central issues in convex geometry. Our aims are the follows. 1) Characterize the Laplace transforms. We try to find the best conditions to characterize the Laplace transform. 2) We want to classify the valuations from the set of convex bodies to the space of continuous functions. These valuations should include Lp centroid body and Lp projection body and the Laplace transform. 3) We will study the dual question of Haberl and Parapatits’s results for measure valued valuations. We expect to characterize the dual Lp surface area measure. 4) We will try to find the Hadwiger’s theorem for bi-valuations. Our aim is to characterize the mix volume. To study the above questions, we will combine the valuations theory with Fourier transform, locally asymptotic convex theory and harmonics analysis. Also we will learn some experience from other mathematicians. By extracting the essential ideas from known methods, we try to develop these methods and find some new methods to attack our aims.