Approximately convex functions

Abstract. A Rolewicz type theorem concerning the superstability of approximate convexity is established. Namely, it is proved that any real valued function \(f\), defined on an open, convex subset \(D\) of a linear normed space, which satisfies the inequality $$ {f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda) f(y) + c \left( \lambda (1-\lambda) \Vert x - y \Vert \right)^p} $$ for every \( x,y \in D \) and \( \lambda \in [0,1] \), with a fixed non-negative real number \(c\), and a fixed exponent \( p > 1 \), has to be convex, i.e., satisfies the above inequality with \( c = 0 \) as well.