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 property means that if you pick any two points in a convex set, you can draw a straight line between them without leaving the set. Convex sets are important in various fields, including mathematics, economics, and optimization. They help in understanding shapes and spaces, making it easier to solve problems related to linear programming and game theory. Examples of convex sets include simple shapes like circles and triangles.