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

Authors: D. Burns, N. Levenberg, S. Ma’u and Sz. Révész
Journal: Trans. Amer. Math. Soc. 362 (2010), 6325-6340
MSC (2010): Primary 32U15; Secondary 41A17, 32W20
Published electronically: July 9, 2010
MathSciNet review: 2678976
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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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Additional Information

D. Burns
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

N. Levenberg
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

S. Ma’u
Affiliation: Mathematics Division, University of the South Pacific, SCIMS, Suva, Fiji

Sz. Révész
Affiliation: A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary

Received by editor(s): May 7, 2007
Received by editor(s) in revised form: July 4, 2008
Published electronically: July 9, 2010
Additional Notes: The first author was supported in part by NSF grants DMS-0514070 and DMS-0805877 (DB)
The fourth author was supported in part by the Hungarian National Foundation for Scientific Research, Project #s K-72731 and K-81658 (SzR)
The third author was supported by a New Zealand Science and Technology Fellowship, contract no. IDNA0401 (SM)
This work was accomplished during the fourth author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927.
Article copyright: © Copyright 2010 American Mathematical Society

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