A Negative Result on Multivariate Convex Approximation by Positive Linear Operators

We consider the problem of multivariate convex approximation by positive linear operators. LetEbe ak-dimensional compact convex set inRkwithk⩾2,Ω⊂Rkan open set containingE, and letL:C(E)→C1(Ω) be a positive linear operator. Our main result of this paper shows that ifLpreserves convexity and satisfiesLl=lonEfor alll∈P1(the space of affine functions), thenLis trivial (i.e.,Lf∈P1onEfor allf∈C(E)) andEis a simplex.