First, let's define what a convex set is. In the real number space R, if all the points connected by line segments are in the set for any two points X and Y in the set, then we call this set a convex set. In other words, if the line between any two points in a set is included in this set, then this set is a convex set.
From a geometric point of view, the shape of a convex set is similar to a "concave" or "U" shape with an upward opening. This is because if we take any two points in the convex set and draw a line segment connecting these two points, all the points on this line segment will be included in the set. This means that we can move in any direction along this line segment without leaving this set. This vividly illustrates the "development" characteristics of convex sets.
In addition, a convex set has another important property, that is, its boundary is also convex. This is because, if the boundary of a set is not convex, then we can find the points on two boundaries, so that the line between them is not on the boundary, which contradicts the definition of convex set. Therefore, all convex sets have a convex boundary.
Generally speaking, the geometric image of a convex set is a "U" shape with an upward opening. Its characteristic is that the connecting line of any two points is included in this set, and its boundary is also convex. These properties make convex sets widely used in mathematical analysis and optimization theory.