  We use geometric methods to calculate a formula for the complex
Monge-Amp\`ere measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ 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 $V_K$.
