Mid-Facets of a Simplex
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摘要: 在n维欧氏空间中，作为三角形的高维推广的单形的中面是最近才引入的一个重要的几何概念．该文利用Grassmann代数的方法获得了单形的中面面积的解析表达式，证明了单形的中面类似于三角形中线的性质，例如，对于一个给定的单形，存在另一个单形使得其边长分别等于给定单形的中面面积；一个单形的所有中面有且仅有一个公共点等．同时，利用中面面积的解析表达式证明了单形中面与单形的棱长、外接圆半径、中线长、角平分面等之间的一些优美性质，建立了一些新的重要的几何不等式．
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关键词:
- 单形 /
- 中面 /
- 中线 /
- Grassmann代数 /
- 几何不等式
Abstract: The mid-facet of a simplex in n-dimensional Euclidean space which was introduced quite recently is an important geometric element. An analytic expression for the mid-facet area of a simplex is firstly given. In order to obtain the expression, the exterior differential method was presented. Furthermore, the properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex, there exists another simplex such that its edge-lengths equal to these mid-facets area respectively, and all of the mid-facets of a simplex have a common point) were proved. Finally, by applying the analytic expression, a number of inequalities which combine edge- lengths, circumradius, median line, bisection area and facet area with the mid-facet area for a simplex were established.-
Key words:
- simplex /
- mid-facet /
- median line /
- Grassmann algebra /
- geometric inequality
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[1] Sanyal A.Medians of a simplex[J].Amer Math Monthly,1967,74(8):967—998. doi: 10.2307/2315279 [2] GUO Shu-guang.Some properties for middle sections of a simplex and their applications[J].J Math(PRC),1997,17(3):413—416. [3] Boothby W M.An Introduction to Differentiable Manifolds and Riemannian Geometry[M].New York:Academic Press,1975. [4] LENG Gang-song,ZHOU Guo-biao.Inverse forms of Hadamard inequality[J].SIAM J Matrix Anal Appl,2002,23(4):990—997. doi: 10.1137/S0895479801387279 [5] Reznikov A G.A strenghened isoperimetric inequality for simplices[A].In:J Lindenstrauss，V D Milmar Eds.Geometric Aspects of Functional Analysis,Lecture Notes in Math[C].1469.New York:Springer,1991,90—93. [6] Schneider R.Covex Bodies: The Brunn-Minkowski Theory[M].Cambridge:Cambridge University Press,1993. [7] LENG Gang-song, SHEN Zhu,TANG Li-hua.Inequalities for two simplices[J].J Math Anal Appl,2000,248(2):429—437. doi: 10.1006/jmaa.2000.6918 [8] Mitrinovic D S,PeAcariAc J E,Volenee V.Recent Advances in Geometric Inequalities[M].Dordrecht, Boston, London:Kluwer Acad Publ,1989. -
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