The Mixed Volume of Two Finite Vector Sets

Abstract

Abstract. We introduce the concept of the mixed volume of two finite vector sets in R n . By employing the exterior differential, we prove a new and powerful inequality and establish a series of quantity relations associated with the mixed volume of two finite vector sets. As applications, we discuss some well-known results of simplices and the Hadamard inequality.

On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions

“…Characterization of the Solutions of the One-Dimensional Identity. When d = 1, the identities of Subsections 3.1 and 3.2 can be reduced to equation (23). That is, f (x) = sin(x) satisfies the functional equation ( 31)…”

On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions

“…Joachimsthal [11] suggested the two-dimensional hypersine (for tetrahedra) and Bartoš [4] extended it to simplices of any dimension. Various authors have explored their properties and applied them to a variety of problems (see e.g., [9,22,13,23,12] and references in there). For our purposes, we have slightly modified the existing definitions, in particular we allow negative values of these functions when the dimension of the ambient space is d + 1.…”