Classically, convex functions are characterized geometrically by the property that the graph is sitting above the tangent lines. Beckenbach replaced the tangent lines by a 2-parameter family of continuous functions, requiring that any two distinct points of the plane can be interpolated by a unique member of the family. Such a 2-parameter family induces a generalized notion of convexity of real-valued functions. In this article, the classical Hermite-Hadamard inequality for convex functions is generalized to the context of generalized convex functions in the sense of Beckenbach.