We use geometric methods to calculate a formula for the complex Monge-Ampere measure (dd~cV_K)~n, for K∈ R~n ∈ C~n a convex body and V_K its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies K. We apply this to show that two methods for deriving Bernstein-Markov type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area ellipses determining the complex Monge-Ampere solution V_K.
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