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

Let X be a linear space,U(?)X be a convex set,and let ? be a nonnegative real number.A function f:U ?R is said to be ?-convex,if it satisfiesf(tx +(1-t)y)? tf(x)+(1-t)f(y)+ ?,for all x,y ?U,t?[0,1].Hyers-Ulam stability theorem states that if X?Rn,nthen for every ?-convex function f defined on the convex set U,there exists a convex function g:U?R and a constant ?(n)>0,such that9(x)? f(x)<g(x)+ ?(n)?,for all x ? U.In 2002,S.J.Dilworth,R.Howard and J.W.Robert further proved the constant?(n)in the stability theorem above can be chosen to beIn this paper,we first prove that ?(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 ?-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.