A convex set is a collection of points in a space where, for any two points within the set, the line segment connecting them also lies entirely within the set. This means that if you pick any two points in the set, you can draw a straight line between them without leaving the set. For example, in a two-dimensional space, a circle or a square is a convex set, while a crescent shape or a star is not. Convex sets are important in various fields, including mathematics, economics, and optimization, as they help simplify problems and ensure certain desirable properties.