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

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 $K$.