| 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）. |
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| The α-Length of Planar Convex Bodies and Isoperimetric Inequalities |
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GUO Huan-huan, LI De-yi, ZOU Du1,2
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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
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| 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 |
