A convex set, in mathematics, refers to a geometric shape or set of points in a vector space where every line segment joining any two points within the set lies entirely within the set itself. More specifically, a subset S of a vector space is considered convex if, for any two points x and y within S, the line segment between them, denoted by [x,y], is also completely contained within S.
Convex sets possess several defining properties. Firstly, the entire set itself is convex, meaning it fulfills the definition of convexity. Additionally, any intersection of convex sets is still convex, and the convex hull of any set preserves the convexity property as well. The boundary of a convex set is smooth and has no sharp corners or indentations.
Convex sets can be visualized as "bulging" or "curved-out" regions where, regardless of the points within the set, the entire region remains within the set. Examples of convex sets include circles, spheres, regular polygons, and closed intervals.
These sets find extensive application in various fields, such as optimization, functional analysis, geometry, and economics. The convexity property provides significant advantages when solving optimization problems, as it ensures the existence of global optima and simplifies the search for optimal solutions.
In summary, a convex set is a geometric shape or set of points where any line segment connecting two points within the set lies entirely within the set itself. They possess properties of smooth boundaries, maintain convexity through intersection or convex hull operations, and find application in diverse mathematical and scientific domains.
