Optimal approximation and quantisation

A number of interesting functionals F(X) of a finite point set X in the Euclidean space can be represented as integrals of a function η that depends on the integration variable y and X restricted onto a certain set C(y,X) that is determined by y and X and satisfies separation and uniform boundedness conditions. For instance, C(y,X) can be the Voronoi cell generated by X that contains y. We single out the general properties of C and η that ensure that the normalised infimum of F(X) over all sets X with cardinality n converges to a limit that can be identified using a particular form of η. The considered functionals include those that appear in quantisation problems for probability measures, and in finding the optimal approximation of a function by splines, tangent planes and triangulated surfaces. For instance, it is shown that the minimum Lβ-approximation error (normalised by nβ) of a sufficiently smooth bivariate convex function f using the convex triangulation converges to the L1/(β+1)-norm of the determinant of the second derivative of f times a certain absolute constant.