Piecewise linear interpolants to Lagrange and Hermite convex scattered data (Q676935)

Consider \(X=(x^1, \dots, x^n)\) to be distinct points in \(\mathbb{R}^2\), and suppose that \(f=(f_1, \dots, f_n)\) are scalar values. Then the Lagrange data set \((X,f)\) is called (strictly) convex, if there exists a (strictly) convex function \(F\) with \(F(x^i) =f_i\) for \(i=1, \dots, n\). Two fundamental interpolants to such data are studied. The first one is the supremum over all convex Lagrange interpolants, which turns out to be piecewise linear on a triangulation. The second one is the infimum over all convex Hermite interpolants and this is piecewise linear on a tessellation. The authors discuss the existence, uniqueness and numerical computation of these two interpolants. Moreover, it is shown how in the strictly convex case Hermite data \((X,f,G)\), where \(G=(g^1, \dots, g^n)\) with \(g^i\in \mathbb{R}^2\) are tangents, may be generated from Lagrange data \((X,f)\) by construction of admissible tangent planes.