MongeAmpère measures for convex bodies and BernsteinMarkov type inequalities
Authors:
D. Burns, N. Levenberg, S. Ma’u and Sz. Révész
Journal:
Trans. Amer. Math. Soc. 362 (2010), 63256340
MSC (2010):
Primary 32U15; Secondary 41A17, 32W20
Published electronically:
July 9, 2010
MathSciNet review:
2678976
Fulltext PDF
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Abstract: We use geometric methods to calculate a formula for the complex MongeAmpère measure , for a convex body and its SiciakZaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies . We apply this to show that two methods for deriving BernsteinMarkov 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 MongeAmpère solution .
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Additional Information
D. Burns
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 481091043
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/S000299472010048925
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 DMS0514070 and DMS0805877 (DB)
The fourth author was supported in part by the Hungarian National Foundation for Scientific Research, Project #s K72731 and K81658 (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 # MEIFCT2005022927.
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© Copyright 2010
American Mathematical Society
