引用本文:郭欢欢, 李德宜, 邹都.平面凸体的α周长及等周不等式[J].西南大学学报(自然科学版),2019,41(10):51~55
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平面凸体的α周长及等周不等式
郭欢欢, 李德宜, 邹都1,2
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
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Lp Brunn-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.
关键词:  凸体  α-周长  Brunn-Minkowski不等式
DOI:10.13718/j.cnki.xdzk.2019.10.007
分类号:O186
基金项目:国家自然科学基金项目(11601399).
The α-Length of Planar Convex Bodies and Isoperimetric Inequalities
GUO Huan-huan, LI De-yi, ZOU Du1,2
1. College of Science, Wuhan University of Science and Technology, Wuhan 430081, China;2. Master Studio of Hubei Province, Wuhan University of Science and Technology, Wuhan 430081, China
Abstract:
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 Lp Brunn-Minkowski theory, the Brunn-minkowski inequality of Lp 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.
Key words:  convex body  α-perimeter  Brunn-Minkowski inequality
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