Interpolation by convex algebraic hypersurfaces
The paper is devoted to the convex algebraic interpolation problem: given a set \(\Sigma\) of vertices of a convex polyhedron in the \(n\)-dimensional Euclidean space, construct a convex algebraic hypersurface of degree as small as possible passing through \(\Sigma\). The main result is that there exists such a hypersurface of degree at most \(d(\Sigma)\), where \(d(\Sigma)\) is the minimal number of simplicial faces of the polyhedron, covering \(\Sigma\). If \(m\) is the number of points in \(\Sigma\), then \(d(\Sigma)\leq(m+1)/2\) in the plane, which implies the Levin-Nadler theorem [\textit{D. Lewin} and \textit{E. Nadler}, Numer. Algorithms 9, No. 1-2, 113-139 (1995; Zbl 0835.41008)], and \(d(\Sigma)<2m/3\) in the three-space. The Hermite-type problem, convex interpolation with given tangent hyperplanes at given points, and the sharpness of upper bounds on \(d(\Sigma)\) are discussed.
scientific article; zbMATH DE number 1006708