Monge-Amp\`ere Measures for Convex Bodies and Bernstein-Markov Type Inequalities

We use geometric methods to calculate a formula for the complex Monge-Amp`ere measure $(d{d}^c{V}_K{)}^n$, for $K⋐{\text{RR}}^n\subset {\text{CC}}^n$ 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-Amp`ere solution $V_K$.