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We discuss the Siciak-Zakharyuta extremal function of pluripotential theory for the unit ball in C d for spaces of polynomials with the notion of degree determined by a convex body P. We then use it to analyze the approximation properties of such polynomial spaces, and how these may differ depending on the function f to be approximated. 2010 Mathematics Subject Classification: Primary 32U15, 41A10.doi:10.4064/ap180322-19-11 fatcat:4qdwsgshgfgvlkrvlazwvtn56m