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. Convex sets are important in various fields, including mathematics, economics, and optimization. Examples of convex sets include simple shapes like circles, squares, and triangles. In contrast, a shape like a crescent moon is not convex because a line segment between certain points would extend outside the shape. Understanding convex sets helps in solving problems related to linear programming and geometry.