无限维\varepsilon-凸函数的Hyers-Ulam逼近

Abstract

设X是线性空间，U是X的凸子集，\varepsilon是非负实数.称函数f:U\rightarrow\mathbb{R}为~$\varepsilon$-凸函数，若对任意x，y\inU，t\in[0，1]，满足 f(tx+(1-t)y)\leq~tf(x)+(1-t)f(y)+\varepsilon. 我们已经知道，若X=\mathbb{R}^n，则对U上任何\varepsilon-凸函数f，存在凸函数g:U\rightarrow\mathbb{R}及常数\kappa(n)>0，使得对任意x\inU，有 g(x)\leqf(x)\leqg(x)+\kappa(n)\varepsilon. ...

Let X be a linear space,\,U\subseteq X be a convex set, and let \varepsilon be a nonnegative real number. A function f:U\longrightarrow \mathbb{R} is said to be \varepsilon-convex, if it satisfies f(tx+(1-t)y)0, such that \begin{align*} &g(x)\leq f(x)\leq g(x)+\kappa(n)\varepsilon, \end{align*} for all x\in U.In 2002, S.J. Robert, R. Howard and J.W. Robert further proved the const...