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 t...

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 2 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 t...

In this paper we consider the following analog of Bezout inequality for mixed volumes: V (P1, . . . , Pr,∆ )Vn(∆) r−1 ≤ r ∏ i=1 V (Pi,∆ ) for 2 ≤ r ≤ n. We show that the above inequality is true when ∆ is an n -dimensional simplex and P1, . . . , Pr are convex bodies in R . We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be an n -dimensional...

In this paper we introduce a new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one quermassintegral due Talenti [26] and Tso [28]. We show Pólya-Szegő type principle for such — it diminishes Hessian integral quasi-convex functions. achieve by systematic study of invariants on non-symmetric matrices real eigenvalues higher order mean curvatu...