Convexity Properties of Some Implicit Functions

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