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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0825731-8
PII: S 0002-9947(1986)0825731-8
Keywords: Plurisubharmonic function, Monge-Ampere operator, extremal function
Article copyright: © Copyright 1986 American Mathematical Society