Monge-ampère Measures for Convex Bodies and Bernstein-markov Type Inequalities

We use geometric methods to calculate a formula for the complex Monge-Ampère measure (ddVK) n, for K Rn ⊂ Cn a convex body and VK 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-Ampère solution VK .