Hyers-Ulam Approximation Of ε-Convex Function On Infinite Dimensional Spaces

Let X be a linear space,U(?)X be a convex set,and let ε be a nonnegative real number.A function f:U →R is said to be ε-convex,if it satisfiesf(tx +(1-t)y)≤ tf(x)+(1-t)f(y)+ ε,for all x,y ∈U,t∈[0,1].Hyers-Ulam stability theorem states that if X＝Rn,nthen for every ε-convex function f defined on the convex set U,there exists a convex function g:U→R and a constant κ(n)>0,such that9(x)≤ f(x)<g(x)+ κ(n)ε,for all x ∈ U.In 2002,S.J.Dilworth,R.Howard and J.W.Robert further proved the constantκ(n)in the stability theorem above can be chosen to beIn this paper,we first prove that κ(n)is the best constant.Then we prove that Hyers-Ulam stability theorem doesn’t hold any more in every infinite dimensional space by con-structing a 1-convex function.Therefore,a linear space X is finite dimensional if and only if every ε-convex function on a convex set satisfies the stability theorem.Finally,we present an approximate Hyers-Ulam stability theorem on infinite dimensional linear spaces and Ba-nach spaces.