Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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

Authors: Eric Bedford and B. A. Taylor
Journal: Trans. Amer. Math. Soc. 294 (1986), 705-717
MSC: Primary 32F05; Secondary 31C10
MathSciNet review: 825731
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $K$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F05, 31C10

Retrieve articles in all journals with MSC: 32F05, 31C10

Additional Information

Keywords: Plurisubharmonic function, Monge-Ampere operator, extremal function
Article copyright: © Copyright 1986 American Mathematical Society