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 | References | Similar Articles | Additional Information

Abstract: We use geometric methods to calculate a formula for the complex Monge-Ampère measure , for a convex body and its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies . 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 .

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Additional Information

**D. Burns**

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

Email:
dburns@umich.edu

**N. Levenberg**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Email:
nlevenbe@indiana.edu

**S. Ma’u**

Affiliation:
Mathematics Division, University of the South Pacific, SCIMS, Suva, Fiji

Email:
mau_s@usp.ac.fj

**Sz. Révész**

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

Email:
revesz@renyi.hu

DOI:
http://dx.doi.org/10.1090/S0002-9947-2010-04892-5

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