等周问题与Bonnesen型Alexandrov-Fenchel不等式研究

The isoperimetric problem is important in Mathematics. The classical isoperimetric problem (CIP) states: The (ellipsoid) ball encloses the maximum volume among compact domain of fixed surface area. The isoperimetric problem is closely related to differential geometry, affine geometry, convex geometry, algebraic geometry, convex geometric analysis, probability, geodesic measure, medical science, atomic physics, information science and other related fields. A special non-negative invariant (which vanishes when the body is the (ellipsoid) ball) of significance of the surface area and volume for convex body K is called the isoperimetric deficit of K. The Bonnesen-type inequality is the natural strengthening and extension. Those generalizations include Minkowsky inequality (for quermassintegrales of convex body)、 Alexandrov-Fenchel inequality (for mixed volumes of convex bodies)、 Gaussian isoperimetric inequality (for distribution function on convex domain). We will investigate those important inequalities and their strengthening forms, that is, the Bonnesen-type Minkowsky inequality、the Bonnesen-type Alexandrov-Fenchel inequality and their dual cases.

等周问题是数学中一个重要问题。古典的等周问题是：具有固定表面积的紧域中（椭）球的体积最大.等周问题涉及微分几何、仿射几何、复几何、代数几何、凸几何分析、概率论、大地测量、医学、核物理、信息科学等领域. 一个与凸（有界闭凸）域的表面积与体积有关的特殊非负量是等周亏格（当凸域为（椭）球时为0），Bonnesen型不等式是等周不等式的加强和推广.等周不等式的自然推广是（关于凸体的均质积分的）Minkowski不等式、（关于 i 个凸体混合体积的）Alexandrov-Fenchel 不等式、（凸域上概率分布的）Gaussian等周不等式等.我们将研究这些重要不等式的加强形式，即Bonnesen型Minkowski不等式、Bonnesen型Alexandrov-Fenchel 不等式等. 我们还将研究这些不等式的“对偶”形式.

我们在等周问题 (Bonnesen-型不等式，逆Bonnesen型对称混合等周不等式;关于两凸域对称混合等周不等式，关于两凸域Bonnesen型对称混合等周不等式，关于两凸域逆Bonnesen型对称混合等周不等式；关于两凸域对称混合Minkowski等似（homothetic）不等式, 关于两凸域对称混合Bonnesen型等似不等式，关于两凸域对称混合逆Bonnesen型等似不等式), Dual Orlicz-Brunn-Minkowski theory,Minkowski问题等方面取得重要进展和结果。特别是Dual Orlicz-Brunn-Minkowski theory上是开创性工作，我们提出了Dual Orlicz-Brunn-Minkowski theory中的“Busemann–Petty problem”，是积分几何与凸几何分析中的重要问题。.在“函数集上的几何”，“函数集上几何不等式”，“Minkowski问题解的连续性”，“(p,q)-John 椭球”，“Isoperimetric inequalities for (p,q)-mixed geominimal surface area and (p,q)-mixed affine surface area”，“The LYZ centroid conjecture for star bodies”上取得突破性进展。