Weights of 2 distribution for smooth or piecewise smooth cone alternatives

We study the problem of testing a simple null hypothesis on multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the  distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of the cone. We illustrate our technique by spherical cone and cone of non-negative de nite matrices.