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

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

Published electronically:
July 9, 2010

MathSciNet review:
2678976

Full-text PDF

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 .

**1.**M. Baran, Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in ,*Michigan Math. J.***39**(1992), 395-404. MR**1182495 (93j:32022)****2.**M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in ,*Proc. AMS***123**(1995), no. 2, 485-494. MR**1219719 (95c:31006)****3.**E. Bedford and B. A. Taylor, The complex equilibrium measure of a symmetric convex set in ,*Trans. AMS***294**(1986), 705-717. MR**825731 (87f:32039)****4.**P. Borwein and T. Erdélyi,*Polynomials and polynomial inequalities*, Graduate Texts in Mathematics, Springer-Verlag, 1995. MR**1367960 (97e:41001)****5.**L. Bos, N. Levenberg and S. Waldron, Pseudometrics, distances, and multivariate polynomial inequalities,*Journal of Approximation Theory***153**(2008), no. 1, 80-96. MR**2432555 (2009d:41015)****6.**D. Burns, N. Levenberg and S. Ma'u, Pluripotential theory for convex bodies in ,*Math. Zeitschrift***250**(2005), no. 1, 91-111. MR**2136404 (2005k:32044)****7.**-, Exterior Monge-Ampère Solutions, Adv. Math.**222**(2009), 331-358. MR**2538012 (2010g:32062)****8.**M. Klimek,*Pluripotential Theory*, Clarendon Press, Oxford, 1991. MR**1150978 (93h:32021)****9.**M. Lundin, The extremal plurisubharmonic function for the complement of the disk in , unpublished preprint, 1984.**10.**-, The extremal plurisubharmonic function for the complement of convex, symmetric subsets of ,*Michigan Math. J.***32**(1985), 197-201. MR**783573 (86h:32030)****11.**L. Milev and Sz. Révész, Bernstein's inequality for multivariate polynomials on the standard simplex,*J. Inequal. Appl.*(2005), no. 2, 145-163. MR**2173358 (2006g:41026)****12.**Sz. Révész, A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials,*Ann. Polon. Math.***88**(2006), no. 3, 229-245. MR**2260403 (2007f:41012)****13.**Sz. Révész and Y. Sarantopoulos, A generalized Minkowski functional with applications in approximation theory,*J. Convex Analysis***11**(2004), no. 2, 303-334. MR**2158907 (2006e:52001)****14.**Y. Sarantopoulos, Bounds on the derivatives of polynomials on Banach spaces,*Math. Proc. Camb. Phil. Soc.***110**(1991), 307-312. MR**1113429 (92j:46084)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
32U15,
41A17,
32W20

Retrieve articles in all journals with MSC (2010): 32U15, 41A17, 32W20

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:
https://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