Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities

Burns, D.; Levenberg, N.; Ma’u, S.; Révész, Sz.

doi:10.1090/S0002-9947-2010-04892-5

Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities
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by D. Burns, N. Levenberg, S. Ma’u and Sz. Révész PDF

Trans. Amer. Math. Soc. 362 (2010), 6325-6340 Request permission

Abstract:

We use geometric methods to calculate a formula for the complex Monge-Ampère measure $(dd^cV_K)^n$, for $K \Subset \mathbb {R}^n \subset \mathbb {C}^n$ 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-Ampère solution $V_K$.

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