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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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
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  • N. Levenberg
  • Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
  • MR Author ID: 113190
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  • S. Ma’u
  • Affiliation: Mathematics Division, University of the South Pacific, SCIMS, Suva, Fiji
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  • Sz. Révész
  • Affiliation: A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary
  • Email:
  • 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.
  • © Copyright 2010 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 6325-6340
  • MSC (2010): Primary 32U15; Secondary 41A17, 32W20
  • DOI:
  • MathSciNet review: 2678976