平面凸体的α-周长及等周不等式  

The α-Length of Planar Convex Bodies and Isoperimetric Inequalities

作　　者：郭欢欢 李德宜[1,2] 邹都[1] GUO Huan-huan;LI De-yi;ZOU Du(College of Science,Wuhan University of Science and Technology,Wuhan 430081,China;Master Studio of Hubei Province,Wuhan University of Science and Technology,Wuhan 430081,China)

机构地区：[1]武汉科技大学理学院,武汉430081 [2]武汉科技大学湖北名师工作室,武汉430081

出　　处：《西南大学学报（自然科学版）》2019年第10期51-55,共5页Journal of Southwest University(Natural Science Edition)

基　　金：国家自然科学基金项目(11601399)

摘　　要：Brunn-Minkowski不等式是凸几何分析的重要研究内容.目前,关于体积等几何量的Brunn-Minkowski不等式已广为人知,并在数学各个分支中扮演着重要的角色.关于凸体表面积的Brunn-Minkowski不等式作为Aleksandrov-Fenchel不等式的特殊情况也得到确证.但在LpBrunn-Minkowski理论中,Lp表面积测度的Brunn-Minkowski不等式仍是一个重要的公开问题,不论是对0<p<1,还是p>1的情形,都没有行之有效的方法来证明相关猜测.基于Minkowski加法,利用单调有界定理和积分中值定理研究了平面凸体的α-周长,提出了两凸体关于α-周长的Brunn-Minkowski型不等式,并对两凸体分别为正n边形和单位圆盘的情形给出了证明.The Brunn-Minkowski inequality is an important research content of convex geometry analysis.At present,the Brunn-minkowski inequality about volume and other geometric quantities is widely known and plays an important role in various branches of mathematics.Brunn-Minkowski inequality of convex body surface area as a special case of Aleksandrov-Fenchel inequality has also been confirmed.But in L p Brunn-Minkowski theory,the Brunn-minkowski inequality of L p surface area measurement is still an important open problem.There is no effective method to prove the related conjecture for 0< p <1 and p >1.In this paper,based on the addition of Minkowski,the monotone bounded theorem and integral mean value theorem are used to study the α-perimeter of convex body in the plane.The Brunn-Minkowski type inequality about α-perimeter is put forward and proved when two convex bodies are a regular n polygon and a unit disc,respectively.

关 键 词：凸体 α-周长 BRUNN-MINKOWSKI不等式 

分 类 号：O186[理学—数学]