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Convex sets

Let be a subset of a real or complex vector space . The is said to be convex if for all and all the point is in . In other words, every point on the line segment connecting and is in .  The empty set is considered as convex.

If the set does not contain all the line segments, it is called concave or non-convex.

If is a convex set then for any and such that , then the point is also in .

Convex hull of a set of a complex or real  vector space is the smallest convex set containing . Since the intersection of any collection of convex sets is itself convex, the convex hull of is the intersection of all convex sets containing .

A set of a complex or real vector space is  called star convex if there exists an such that all line segments from to any point are contained in . A convex set is always star convex but the converse is not true in general.

There are numerous geometry algorithms for computing of convex hulls in the plane. For the most common of these algorithms visit   .

The convexity notion may be generalizes to other type of spaces or structures in such a way that the following properties are fulfilled:

• The empty set is convex.
• The intersection of any collection of convex sets is again a convex set.
• The union of a chain of sets ordered with respect to the set inclusion is again a convex set.

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