The complex equilibrium measure of a symmetric convex set in 𝑅ⁿ

Source abstract

We give a formula for the measure on a convex symmetric set K K in R n {{\mathbf {R}}^n} which is the Monge-Ampere operator applied to the extremal plurisubharmonic function L K {L_K} for the convex set. The measure is concentrated on the set K 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 ∈ K x \in K near a boundary point x 0 {x_0} of K K is shown to be of the form c ( x 0 ) / [ dist ( x , ∂ K ) ] − 1 / 2 c({x_0})/{[{\operatorname {dist}}(x,\,\partial K)]^{ - 1/2}} , where the constant c ( x 0 ) c({x_0}) depends both on the curvature of K K at x 0 {x_0} and on the global structure of K K .