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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The complex equilibrium measure of a symmetric convex set in $\textbf {R}^ n$
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by Eric Bedford and B. A. Taylor PDF
Trans. Amer. Math. Soc. 294 (1986), 705-717 Request permission

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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Additional Information
  • © Copyright 1986 American Mathematical Society
  • 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