Convex analysis is a branch of mathematics that focuses on the study of convex sets and convex functions. A set is considered convex if, for any two points within the set, the line segment connecting them also lies entirely within the set. Convex functions, on the other hand, have the property that a line segment connecting any two points on the graph of the function lies above or on the graph itself. This field has important applications in various areas, including optimization, economics, and machine learning. By understanding the properties of convexity, researchers can develop efficient algorithms for solving problems where finding the best solution is crucial.