证明了En中的有序向量集是伪对称点集的充要条件. 利用这一充分必要条件,得到了有关正则单形的几个等价描述,给出了伪对称点集与正则单形的关系的一个结论:设Im={A1,A2,…,An+1}是En中的点集, 则Im是n维对称点集的充要条件是以Im为顶点的单形是正则单形.
Abstract
A sufficient and necessary condition that an ordered vectors-set is a pseudosymmetric point set in En is explored. Based on this condition, several equivalent ways of characterizing regular simplices are obtained. A relationship between a pseudosymmetric point set and a regular simplex is given, i.e., let Im={A1,A2,…,An+1}be a point set in En, then Im is an n-pseudo-symmetric point set if and only if the simplex with vertex set Im is regular.
关键词
伪对称点集 /
正则单形 /
Legendre 椭球
{{custom_keyword}} /
Key words
pseudo-symmetric point set /
mass-point system /
Legendre ellipsoid
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Yang L., Zhang J., A sufficient and necessary condition for embedding a simplex with prescribed dihedralangles in En, Acta Mathematica Sinica, Chinese Series, 1983, 26(2): 250-256.
[2] Yang L., Zhang J., Application of metric eqution to Sallee’s guess, Acta Mathematica Sinica, Chinese Series,1983, 26(2): 488-493.
[3] Ali M. M., On some extremal simplexes, Pacific J. Math., 1970, 33: 1-14.
[4] Gerber L., The orthocentric simplex as an extreme simplex, Pacific J. Math., 1975, 56: 97-111.
[5] Petty C. M., Waterman D., An extremal theorem for n-simplexes, Monatsh Math., 1955, 59: 320-322.
[6] Slepian D., The content of some extreme simplexes, Pacific J. Math., 1969, 31: 795-808.
[7] Yang L., Zhang J., Pseudo-symmetric point set and geometric inequalities, Acta Mathematica Sinica, ChineseSeries, 1986, 29(6): 802-806.
[8] Coxeter H. S. M., Regular Polytopes, Courier Dover, New York, 1973.
[9] Filliman P., The extreme projections of the regular simplex, Tran. Amer. Math. Soc., 1990, 317(2): 611-629.
[10] Kawashima T., Polytopes which are orthogonal projections of regular simplices, Geom. Dedicata, 1991, 38:73-85.
[11] Kawashima T., On a theorem of Kurnik, Glasnik Mat., 1989, 24(44): 77-88.
[12] Arnold V. I., Mathematical Methods of Classical Mechanics, (Translated by K. Vogtm and A. Weinstein),Springer-Verlag, New York, 1978.
[13] Klamkin M. S., Geometric inequalities via the polar moment of inertia, Math. Mag., 1975, 48: 44-46.
[14] Leng G., Zhang Y., The generalized sine theorem and inequalities for simplices, Linear Algebra and its Appl.,1998, 278: 237-247.
[15] Yang L., Zhang J., A class of geometric inequalities concerning the mass-point systems, J. China Univ. Sci.Technol., Chinese Series, 1981, 24(2): 1-8.
[16] Zhou J., Sufficient and necessary condition for a pseudo-symmetric point set, J. Math. Res. Exposition, 1990,10: 65-68.
[17] Mullen P. M., Volumes of projections of unit cubes, Bull. London Math. Soc., 1984, 16: 278-280.
[18] YANG S. G., Two inequalities for pedal simplex and applications, Journal of Zhejiang Univesity (ScienceEdition), Chinese Series, 2005, 32(6): 621-623.
[19] Leng G., Qian X., Inequalities for any point and two simplices, Discrete Math., 1999, 202: 163-172.
[20] Zhang Y., A conjecture for the volume of pedal simplices, J. System Sci. Math. Sci., Chinese Series, 1992,12(4): 371-375.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
基金
国家自然科学基金资助项目(10971128)
{{custom_fund}}
PDF(429 KB)
