E^n的广义欧拉不等式(英文)

Generalizations of Euler Inequality in E^n

用R和r表示n维欧氏空间中的n维单形Ω的外接超球半径和内切超球半径 ,给出了欧拉不等式R ≥nr的广义形式 :R2 ≥ (ur) 2sinθ +｜OI｜2 ,R2 ≥ (nr) 2sinθ +｜OG｜2 ,其中O ,I和G分别是单形Ω的外心、内心和重心 ,θ是单形Ω所有相对棱夹角的平均值 ,等号当且仅当单形Ω正则时成立 .

Denote by R and r the circumradius and the in radius of a n-dimensional simplex Ω in the n-dimensional Euclid ian space E-n. The Euler inequality R≥nr is generalized as R-2≥(nr)-2/sinθ+｜OI｜-2. R-2≥(nr)-2/sinθ+｜OG｜-2, where O,I and G are the circumcenter, incenter and barycenter of Ω respectively and θ is the arithmetic mean value of all angles of opposite edges of Ω, and the equality occur if and only if Ω is regul ar.