MONGE-AMPERE MEASURES FOR CONVEX BODIES AND BERNSTEIN-MARKOV TYPE INEQUALITIES

We use geometric methods to calculate a formula for the complex Monge-Ampere measure (dd(e)V(K))(n), for K is an element of R(n) subset of C(n) a convex body and V(K) 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).