Discrete and Continuous Bang-Bang and Facial Spaces Or: Look for the Extreme Points

A simple geometrical argument is used to establish seemingly different continuous and discrete hang-hang type results. Among other applications we discuss the bang-bang principle for linear continuous control systems, a generalization to discrete systems, the ranges of vector integrals, the Shapley–Folkman lemma and the Carathéodory theorem, a moment problem, and a minimum norm problem.