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

TL;DR

This paper derives a formula for the complex Monge-Ampère measure of convex bodies using geometric methods and shows that two different approaches to Bernstein-Markov inequalities are equivalent for all convex bodies, highlighting the role of extremal inscribed ellipses.

Contribution

It provides a new geometric formula for the Monge-Ampère measure of convex bodies and proves the equivalence of two Bernstein-Markov inequality methods for all convex bodies.

Findings

Derived a formula for the Monge-Ampère measure $(dd^cV_K)^n$ for convex bodies.

Proved the equivalence of two Bernstein-Markov inequality methods.

Identified extremal inscribed ellipses as maximal area ellipses in approximation theory.

Abstract

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 $V_K$.