ON STABILITY OF A CLASS OF CONVEX FUNCTIONS

Abstract:

We study stability problems of the class ℨ_co(G) of all convex functions g : Δ ⊂ ℝⁿ → ℝ defined on domains Δ ⊂ ℝⁿ and satisfying the differential inclusion

In essence, stability means that local proximity of a mapping f to the mappings of the class ℨ_co(G) implies global proximity of f to them in the C-norm. We prove that the class ℨ_co(G) is stable if the projection of the compact G on each straight line l ⊂ ℝⁿ is a totally disconnected set. The latter condition is also necessary in the case of functions of one variable.