Convex

Convex is a term used in geometry and mathematics to describe a shape or figure that has no inward curves or depressions. In a convex figure, all interior angles are less than or equal to 180 degrees and every line segment connecting two points in the figure is entirely contained within the figure.

Definition

In mathematical analysis, a set of points in a real or complex vector space is said to be convex if, given any two points in the set, the set contains the whole line segment that joins them. This definition can be extended to include sets in n-dimensional space.

Properties

Convex sets have several important properties. They are always connected and closed, meaning they contain all their limit points. They are also always bounded, meaning they can be contained within a finite volume. Additionally, the intersection of any collection of convex sets is itself convex.

Applications

Convexity has applications in a variety of fields, including computer science, economics, and optimization theory. In computer science, for example, convex hull algorithms are used in image processing and machine learning. In economics, convexity is used in the study of consumer choice and the theory of the firm.

See also

• Concave
• Convex function
• Convex hull
• Convex optimization
• Convex polygon
• Convex set

References

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