THE COMPLEX EQUILIBRIUM MEASURE OF A SYMMETRIzC CONVEX SET IN Rn

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 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 E K near a boundary point xo of K is shown to be of the form c(xo)/[dist(x,8K)]-1/2, where the constant c(xo) depends both on the curvature of Kat xo and on the global structure of K.