Barycentric coordinates and the orientation of the classical mixture surface

The signs of the barycentric coordinates of a point exterior to a nondegenerate k-simplex in IR^P contain useful information about how that point is positioned relative to the vertices of that simplex. This relationship is certainly not newly observed, with some of the first ideas dating back to Möbius in 1827. However, this article presents some new geometrical results which further quantify the relationship and focuses on applying these new results to help solve the problem of finding the point on a simplex that is closest to a given exterior point. In particular, it is shown that the signs of the barycentrics can be used to immediately identify a potentially large set of facets that could not contain this closest point. Such results have immediate applications to the poblem of identifying the components in a chemical linear mixture. Real PCB mixtures are employed to illustrate the new ideas.