Set-Valued Analysis of Generalized Barycentric Coordinates and Their Geometric Properties

Letting $P$ be a convex polytope in ${\mathbb{R}}^d$ with $n>d$ vertices, we study geometric and analytical properties of the set of generalized barycentric coordinates relative to any point $p\in P$. We prove that such sets are polytopes in ${\mathbb{R}}^n$ with at most $n-d-1$ vertices, and provide results about continuity and differentiability for the corresponding set-valued maps.