
Journal of Convex Analysis 08 (2001), No. 1, 241254 Copyright Heldermann Verlag 2001 Convexity Properties of Some Implicit Functions U. Würker Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany 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. Keywords: Convex function, implicit function, convex parametric optimization. MSC: 26B10; 26B25, 52A20, 90C25. [ Fulltextpdf (293 KB)] 