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Transactions of the American Mathematical Society

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The complex equilibrium measure of a symmetric convex set in $ {\bf R}\sp n$


Authors: Eric Bedford and B. A. Taylor
Journal: Trans. Amer. Math. Soc. 294 (1986), 705-717
MSC: Primary 32F05; Secondary 31C10
DOI: https://doi.org/10.1090/S0002-9947-1986-0825731-8
MathSciNet review: 825731
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Abstract: We give a formula for the measure on a convex symmetric set $ K$ in $ {{\mathbf{R}}^n}$ which is the Monge-Ampere operator applied to the extremal plurisubharmonic function $ {L_K}$ for the convex set. The measure is concentrated on the set $ K$ and is absolutely continuous with respect to Lebesgue measure with a density which behaves at the boundary like the reciprocal of the square root of the distance to the boundary. The precise asymptotic formula for $ x \in K$ near a boundary point $ {x_0}$ of $ K$ is shown to be of the form $ c({x_0})/{[{\operatorname{dist}}(x,\,\partial K)]^{ - 1/2}}$, where the constant $ c({x_0})$ depends both on the curvature of $ K$ at $ {x_0}$ and on the global structure of $ K$.


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  • [1] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Invent. Math. 37 (1976), 1-44. MR 0445006 (56:3351)
  • [2] -, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40. MR 674165 (84d:32024)
  • [3] -, Fine topology, Šilov boundary, and $ {(d{d^c})^n}$, preprint.
  • [4] -, On the convex hull of ellipsoids, preprint.
  • [5] N. Levenberg, Monge-Ampère measures associated to extremal plurisubharmonic functions in $ {{\mathbf{C}}^n}$, Trans. Amer. Math. Soc. 289 (1985), 333-343. MR 779067 (86i:32030)
  • [6] M. Lundin, The extremal plurisubharmonic function for convex symmetric subsets of $ {{\mathbf{R}}^n}$, Michigan Math. J. 32 (1985), 197-201. MR 783573 (86h:32030)
  • [7] -, An explicit solution to the complex Monge-Ampere equation, preprint.
  • [8] Nguyen Thanh Van and A. Zeriahi, Familles de polynomes presque partout bornées, Bull. Sci. Math. 197 (1983), 81-91. MR 699992 (85b:32026)
  • [9] W. Plesniak, Sur la $ L$-regularité des compacts de $ {{\mathbf{C}}^n}$, Séminaire d'Analyse Complex de Toulouse.
  • [10] -, $ L$-regularity of subanalytic sets in $ {{\mathbf{R}}^n}$, preprint.
  • [11] J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampere equation, Rocky Mountain J. Math. 7 (1977), 345-364. MR 0454331 (56:12582)
  • [12] A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys 36 (1981), 61-119. MR 629683 (83c:32026)
  • [13] J. Siciak, Extremal plurisubharmonic functions in $ {{\mathbf{C}}^n}$, Proc. 1st Finnish-Polish Summer School in Complex Analysis, 1977, pp. 115-152. MR 0590080 (58:28670)
  • [14] -, On some inequalities for polynomials, Zeszty Nauk. Uniw. Jagielloń 21 (1979), 7-10.
  • [15] -, Extremal plurisubharmonic functions and capacities in $ {{\mathbf{C}}^n}$, Sophia Kokyuroku Math. 14, Sofia Univ., Tokyo, 1982.
  • [16] B. A. Taylor, An estimate for an extremal plurisubharmonic function in $ {{\mathbf{C}}^n}$, Séminaire d'Analyse, P. Lelong-P. Dolbeault-H. Skoda, 1982-1983, Lecture Notes in Math., vol. 1028, Springer, 1983, pp. 318-328. MR 774982 (86g:32025)
  • [17] V. Zaharjuta, Orthogonal polynomials, and the Bernstein-Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math. 33 (1976), 137-148. MR 0444988 (56:3333)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0825731-8
Keywords: Plurisubharmonic function, Monge-Ampere operator, extremal function
Article copyright: © Copyright 1986 American Mathematical Society

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