Christoffel functions, $$L^p$$ Markov inequalities and Marcinkiewicz-Zygmund type discretization for homogeneous polynomials on convex bodies

Abstract In this paper we will study various extremal problems related to multivariate homogeneous polynomials on convex bodies. We will construct certain homogeneous needle polynomials on convex bodies which will play a central role in the study of the asymptotics of Christoffel functions and $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> Markov type estimates for homogeneous polynomials. These new results will be applied in order to derive asymptotically optimal Marcinkiewicz-Zygmund type meshes and cubature formulas for homogeneous polynomials on convex bodies