多元凸函数连续性的2种证明  

Two Proofs of the Continuity of Multivariate Convex Functions

作　　者：唐建国 叶桂余 TANG Jianguo;YE Guiyu(School of Mathematics and Statistics,Huizhou University,Huizhou 516007,Guangdong,China)

机构地区：[1]惠州学院数学与统计学院,广东惠州516007

出　　处：《惠州学院学报》2020年第6期1-7,共7页Journal of Huizhou University

基　　金：广东省人才引进资金项目(A410.0204)。

摘　　要：文章给出了n维欧氏空间中开凸集上的n元凸函数是连续函数的2种证明方法.首先利用开凸集中长方体内任意一点可由其顶点的线性组合表示,给出了n元凸函数连续性的证明.然后给出了n元凸函数定义的一个等价形式,证明了沿经过开凸集内任意一点直线的两个方向导数都存在,并由此给出了开凸集上n元凸函数连续性的另一证明.we devise two methods to prove that the n-variable convex function on an open convex set in n-dimension Euclidean space is a continuous function.We first prove the continuity of the n-variable convex function by using the linear combination of the verticals of any point in the cuboid of the open convex set.Then an equivalent form of the definition of n-variable convex function is given.It is proved that there are two directional derivatives along a straight line passing through any point in the open convex set.Furthermore,the continuity of the n-variable convex functions on the open convex set is proved in another method.

关 键 词：多元凸函数 连续性 内点表示证明法 方向导数证明法 

分 类 号：O172.1[理学—数学]