## 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 $V_K$.## References

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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
- MR Author ID: 113190
- 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
- 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: https://doi.org/10.1090/S0002-9947-2010-04892-5
- MathSciNet review: 2678976