The interpolation problem asks whether a given configuration of points in space lies on a geometric object with prescribed properties. This question arises naturally in a variety of contexts, from the geometry of data sets to classical problems in algebraic geometry. In this talk, we focus on configurations of points in projective space and ask when they lie on a rational normal curve. Our main goal is to generalize to higher dimensions some classical results of Brianchon and von Staudt concerning the existence of such curves through specific point configurations. The original results presented are in collaboration with Alessio Caminata and Enrico Carlini.