摘要
§1 引言全文约定 k（k=2,3,…,）维欧氏空间 E<sup>k</sup> 中 k 维单形Ω（A<sub>k</sub>）的顶点集为 A<sub>k</sub>={P<sub>0</sub>,P<sub>1</sub>,…,P<sub>k</sub>},棱长为■=a<sub>ij</sub>(i,j=0,1,…k;a<sub>ij</sub>=a（ji）,a<sub>ij</sub>=0),外接超球的半径为R<sub>k</sub>,体积为 V<sub>k</sub>,诸棱长的积为 P<sub>k</sub>=multiply from 0【i【j【n a<sub>ij</sub>.在1974年 Korchmaros G.在[1]中证明了 Veljan D.于1970年提出的如下猜想:对 E<sup>n</sup> 中 n 维单形Ω（A<sub>n</sub>）。
In this paper,Veljan-Korchmaros inequality in [1] is improved and thefollowing theorem is proved.Theorem Let Ω(A_n)={p_0,p_1,…,P_n} is a n dimensional simplex in E^n,a_(ij)=|(?)|(i≠j,i,j=0,1,…,n;a_(ij)=a_(ji) denote its edge lengths.V_n and R_ndenote its volume and circumradius,then (1)V_n≤((n+1)^(1/2))/(?) (multiply from 0≤i<j≤n a_(ij))^(2/(n+1))H_nand the equality holds if and only if Ω(A_n) is a regular simplex,whereH_2=[1-(α_(01)-α_(02))~2(α_(02)-α_(02))~2(α_(12)-α_(02))/(α_(01)α_(02)α_(12))]^(?)H_n={1-(?) [(α_(ij)α_(rs)-α_(ir)α_(js))~2(α_(ir)α_(js)-α_(is)α_(jr))~2.·(a_(is)a_(jr)-a_(ij)a_(rs))~2/(a_(ij)a_(ir)a_(is)a_(jr)a_(js)a_(rs))~2]}^(1/4)(n≥3).(2)V_nR_n≤(n^(1/2))/(?)(multiply from i< ≤a_(ij))^(2/n) K_n,and the equality holds if and only if there exist n+1 positive numbers c_(?)(i==0,1,2,…,n) such thata_(ij)=c_(i)c_(j)(i≠j,i,j=0,1,…,n),where K_2=1,K_3=H_(3^(2/3)),K_n=H_n(n≥4).(3)for any real number μ_i(i=0,1,…,n,sum from i=0 to n μ_i^2≠0),the following inequa-lity holds(sum from 0≤i<j≤n μ_iμ_jα_(ij)~2)V_n^2≤n/(2(n+1)((?))~2)(sum from i=0 to n μ_i)~2(multiply from 0≤i<j≤n α_(ij))^(4/n) K_n^2,where K_2=1,K_3=H_3^(2/3),K_n=H_n(n≥4),and equality holds if and only ifsum from i=0 to n μ_i≠0 and volume coordinate of circumcenter of Ω(A_n) is (μ_0∶μ_1∶μ…∶μ_n)and there exsit n+1 positive numbers c_i(i=0,1,2,…,n) such thata_(ij)=C_iC_j(i≠j,i,j=0,1,…,n).
出处
《数学杂志》
CSCD
北大核心
1990年第4期413-420,共8页
Journal of Mathematics
