欧氏空间中闭曲线的切线象

THE TANGENT SPHERICAL IMAGE OF A COLOSED CURVE IN THE EUCLIDEAN SPACE

W.Fenchel曾于1928年证明:3维欧氏空间中光滑闭曲线的切线象的长不小于2π在本文中我们证明了下述定理;定理 设c’是n维欧氏空间中分段光滑闭曲线c的切线象,则必存在一个内接于c’的球面m边形(m≤n+1),其长不小于2π.它是Fenchel定理的推广.

W. Fenchel proved that the lehgth of the tangent spherical image of a closed curve in three-dime-nional Euclidean space is not less than 2n. The following theorem, a generization of Fenchel's, is proved :Let c be a piecewise smooth closed curve in a w-dimensional Euclidean space and d the tangent spherical image of c. Then there is a spherical wi-polygon (m<n=1) p inscribed in d and the length of p is not less than 2n.