I am studying now convex sets and very interested in applications of them in Computer Science (maybe ACM problems) and other real life problems. Coud you please give some examples?
P.S. I am interested in examples that are strictly related to convex sets. And without this notion they would not be solved.
Peter comment answer pretty well this question. It can be found misplaced at What are the main uses of Convex Functions? by Evgeny adding uniqueness. So, in my oppinion optimization is the key:
Every local extremum is also a global extremum and there exists only one such extremum.