A compact convex set is a type of mathematical object found in geometry and analysis. It is defined as a set that is both compact, meaning it is closed and bounded, and convex, meaning that for any two points within the set, the line segment connecting them also lies entirely within the set. These sets are important in various fields, including optimization and economics, as they often represent feasible regions for problems. Examples of compact convex sets include closed intervals in one-dimensional space and solid shapes like cubes or spheres in higher dimensions, which are often studied in relation to convex analysis and topology.