Motivated by the classification of finite-dimensional, complex, semisimple Lie algebras, the theory of Kac-Moody groups & algebras provides a natural generalization of classical Lie theory to an infinite-dimensional setting. Parallel to the finite case, Kac-Moody algebras are associated with rigid combinatorial structures, such as root systems and Weyl groups, but in contrast, these objects are infinite in the general Kac-Moody setting. The goal of this talk is to present a result of my PhD-project that naturally leads to studying convex hulls of Weyl group orbits associated to Kac-Moody algebras and understanding their facial structure. Although convex hulls of infinite sets usually appear to be rather wild objects and there hardly exists any general theory, I will explain how faces of convex hulls of Weyl group orbits can be explicitly constructed and described via the powerful theory of Coxeter groups. The talk is intended to not require any prior knowledge of Coxeter groups and Lie theory, they will mainly serve as a motivational perspective. In fact, the problem that I will present is mostly a matter of linear algebra.