Approximations of differentiable convex functions on arbitrary convex polytopes

Let Xn≔{xi}i=0n be a given set of (n+1) pairwise distinct points in Rd (called nodes or sample points), let P=conv(Xn), let f be a convex function with Lipschitz continuous gradient on P and λ≔{λi}i=0n be a set of barycentric coordinates with respect to the point set Xn. We analyze the error estimate between f and its barycentric approximation:Bn[f](x)=∑i=0nλi(x)f(xi),(x∈P)and present the best possible pointwise error estimates of f. Additionally, we describe the optimal barycentric coordinates that provide the best operator Bn for approximating f by Bn[f]. We show that the set of (linear finite element) barycentric coordinates generated by the Delaunay triangulation gives access to efficient algorithms for computing optimal approximations. Finally, numerical examples are used to show the success of the method.