Geometric computation of Christoffel functions on planar convex domains

Abstract

For an arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelogram containing the domain is constructed.As an application we obtain a new proof that every planar convex domain possesses optimal polynomial meshes.