A real-valued function f defined on a convex subset D of some normed linear space X is said to be inner y-convex w.r.t. some fixed roughness degree gamma 0 if there is a v belong [0, 1] such that sup(f((l-A)x0 + Ax1) - (1 - A)f(xo) - A(x1)) or = 0 holds for all xo, x1 belong D satisfying ||xo - x1|| = vy and -(1/v)xo + (1 + l/v)x1 belong D. The requirement of this kind of roughly generalized convex functions is very weak
nevertheless, they also possess properties similar to those of convex functions relative to their supremum. For instance, if an inner gamma-convex function defined on some bounded convex subset D of an inner product space attains its maximum, then it has maximizers at some strictly gamma-extreme points of D. In this paper, some sufficient conditions and examples for gamma-convex functions and several properties relative to the location of their maximizers are given.