Probability of Random Vector Hitting Truncated Polyhedral Cone: Majorization Aspect

The paper presents conditions under which the probability of hitting a linear combination of random vectors in a compressed (from above) polyhedral cone, in particular, a truncated cone is a Schur-concave function of the vector corresponding to this linear combination. It is required that the compressed cone be convex, contain the point 0, its edges be parallel to the coordinate axes, and the distribution density of the vectors be a logarithmically concave sign-invariant function. In addition, a characterization of functions that preserve one known preorder inside the majorization preorder is obtained in differential form.