On approximately convex functions

The Bernstein-Doetsch theorem on midconvex functions is extended to approximately midconvex functions and to approximately Wright convex functions. Let X be a real vector space, D be a convex subset of X, and c be a nonnegative constant. A function f: D -R is said to be e-convex if f(tx + (1 t)y) 0 such that tx + (1 t)xo E A for all t E ] , 3[. In earlier literature such sets A are called algebraically open [11] (cf. also [3-5, 7]). Lemma 1. If D is convex and f: D -R is c-midconvex, then (1) f(k2-nx + (1 k2-n)y) < k2-nf(x) + (1 k2-n)f(y) + (2 -2-n+,) for all x, y E D, n E N = {l , 2, ... }, and k E {0, 1, ..., 2n}. Received by the editors August 8, 1991; presented to the 96th Annual Meeting of the AMS on January 19, 1990. 1991 Mathematics Subject Classification. Primary 26B25, 26A5 1; Secondary 39B72.