The complex equilibrium measure of a symmetric convex set in ⁿ

We give a formula for the measure on a convex symmetric set K in Rn which is the Monge-Ampere operator applied to the extremal plurisub- harmonic function LK 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 E K near a boundary point xo of K is shown to be of the form c(xo)/(dist(x, AK)) -1/2, where the constant c(xo) depends both on the curvature of K at xo and on the global structure of K.