Guessab, Allal Generalized barycentric coordinates and Jensen type inequalities on convex polytopes. (English) Zbl 1339.41031 J. Nonlinear Convex Anal. 17, No. 3, 527-547 (2016). Summary: In this paper, we obtain some direct and converse new multidimensional Jensen’s type inequalities on convex polytopes. Among the inequalities presented, we offer, as a particular case of our general results, a direct and converse multivariate extension of Mercer inequality. The main results are obtained with the help of the generalized barycentric coordinates. For deriving such inequalities, we will also establish, analyze, and discuss links between barycentric coordinates and certain class of partitions of unity. This method also allows us to derive continuous versions of various discrete inequalities established in our recent paper [the author, J. Nonlinear Convex Anal. 13, No. 4, 777–797 (2012; Zbl 1259.41032)]. Cited in 1 Document MSC: 41A36 Approximation by positive operators 41A63 Multidimensional problems 41A80 Remainders in approximation formulas 47A58 Linear operator approximation theory Keywords:convex functions; Jensen’s inequality; barycentric coordinates; Farkas’ lemma; partitions of unity; supporting hyperplane Citations:Zbl 1259.41032 PDFBibTeX XMLCite \textit{A. Guessab}, J. Nonlinear Convex Anal. 17, No. 3, 527--547 (2016; Zbl 1339.41031) Full Text: Link