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. Examples of convex sets include simple shapes like circles, squares, and triangles. In contrast, a non-convex set contains at least one pair of points where the connecting line segment falls outside the set. Understanding convex sets is important in various fields, including mathematics, economics, and optimization, as they often simplify problems and lead to more efficient solutions.