平面上的逆Bonnesen-型Minkowski不等式

Reverse Bonnesen-Style Minkowski Inequalities in the Plane

研究了平面中形如A^2_(K,L)-A_KA_L≤U_(K,L)的Minkowski不等式的上界,即逆Bonnesen-型Minkowski不等式.设K,L是平面中的凸体,其面积分别为A_K,A_L,其中A_(K,L)为两凸体的混合面积,U_(K,L)为与K,L有关的几何不变量.利用平面上给定两凸体的支持函数,构造一类与给定凸体相关的新凸体.通过对新凸体几何性质的讨论,得到了一些新的加强的逆Bonnesen-型Minkowski不等式,并用此类不等式可推出一些已有结果.

We study in this paper the upper bound of the Minkowski inequality in the plane,i.e.a reverse Bonnesen-style Minkowski inequality,such as A2K,L-A KAL≤UK,L.Let K and L be convex bodies whose areas are AK and AL,respectively,and AK,L is the mixes area of the two convex bodies and UK,L is the geometric invariant related to K and L.We construct a class of convex body by the support function of the given convex bodies.By discussing the geometric properties of the new convex body,we obtain some new stronger reverse Bonnesen-style Minkowski inequalities and some results can be derived from those inequalities.