Convexity Properties of Some Implicit Functions

We consider implicit functions y = y(x) defined by a system of equations $G_i(x,y) = 0$, i=1,...,m. In the case of convex differentiable functions $G_i$ we establish some sufficient conditions under which the component function $y_k(x)$ is convex or concave. Examples show that without these assumptions $y_k(x)$ can be nonconvex and nonconcave. For the special case with additive separated convex functions $G_i(x,y) = g_i(x) + h_i(y)$ additional results concerning the gradient vectors of $g_i$ and $h_i$ are obtained which can be applied to the differentiable continuation of convex marginal functions in parametric optimization.