Editing Convex set (section)

== Generalizations and extensions for convexity ==
The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name "generalized convexity" is used, because the resulting objects retain certain properties of convex sets.

=== Star-convex (star-shaped) sets ===
{{main|Star domain}}
Let {{mvar|C}} be a set in a real or complex vector space. {{mvar|C}} is '''star convex (star-shaped)''' if there exists an {{math|''x''<sub>0</sub>}} in {{mvar|C}} such that the line segment from {{math|''x''<sub>0</sub>}} to any point {{mvar|y}} in {{mvar|C}} is contained in {{mvar|C}}. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.

=== Orthogonal convexity ===
{{main|Orthogonal convex hull}}
An example of generalized convexity is '''orthogonal convexity'''.<ref>Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations",  in: ''Computational Morphology'', 137-152. [[Elsevier]], 1988.</ref>

A set {{mvar|S}} in the Euclidean space is called '''orthogonally convex''' or '''ortho-convex''', if any segment parallel to any of the coordinate axes connecting two points of {{mvar|S}} lies totally within {{mvar|S}}. It is easy to prove that an intersection of any collection of orthoconvex sets is orthoconvex. Some other properties of convex sets are  valid as well.

=== Non-Euclidean geometry ===
The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a [[geodesic convexity|geodesically convex set]] to be one that contains the [[geodesic]]s joining any two points in the set.

=== Order topology ===
Convexity can be extended for a [[totally ordered set]] {{mvar|X}} endowed with the [[order topology]].<ref>[[James Munkres|Munkres, James]]; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). {{ISBN|0-13-181629-2}}.</ref>

Let {{math|''Y'' ⊆ ''X''}}. The subspace {{mvar|Y}} is a convex set if for each pair of points {{math|''a'', ''b''}} in {{mvar|Y}} such that {{math|''a'' ≤ ''b''}}, the interval {{math|[''a'', ''b''] {{=}} {''x'' ∈ ''X'' {{!}} ''a'' ≤ ''x'' ≤ ''b''} }} is contained in {{mvar|Y}}. That is, {{mvar|Y}} is convex if and only if for all {{math|''a'', ''b''}} in {{mvar|Y}}, {{math|''a'' ≤ ''b''}} implies {{math|[''a'', ''b''] ⊆ ''Y''}}.

A convex set is {{em|not}} connected in general: a counter-example is given by the subspace {1,2,3} in {{math|'''Z'''}}, which is both convex and not connected.

=== Convexity spaces ===
The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as [[axiom]]s.

Given a set {{mvar|X}}, a '''convexity''' over {{mvar|X}} is a collection {{math|''𝒞''}} of subsets of {{mvar|X}} satisfying the following axioms:<ref name="Soltan"/><ref name="Singer"/><ref name="vanDeVel" >{{cite book|last=van De Vel|first=Marcel L. J.|title=Theory of convex structures|series=North-Holland Mathematical Library|publisher=North-Holland Publishing Co.|location=Amsterdam|year= 1993|pages=xvi+540|isbn=0-444-81505-8|mr=1234493}}</ref>

#The empty set and {{mvar|X}} are in {{math|''𝒞''}}.
#The intersection of any collection from {{math|''𝒞''}} is in {{math|''𝒞''}}.
#The union of a [[Total order|chain]] (with respect to the [[inclusion relation]]) of elements of {{math|''𝒞''}} is in {{math|''𝒞''}}.

The elements of {{math|''𝒞''}} are called convex sets and the pair {{math|(''X'', ''𝒞'')}} is called a '''convexity space'''. For the ordinary convexity, the first two axioms hold, and the third one is trivial.

For an alternative definition of abstract convexity, more suited to [[discrete geometry]], see the ''convex geometries'' associated with [[antimatroid]]s.

=== Convex spaces ===
{{main|Convex space}}

Convexity can be generalised as an abstract algebraic structure: a space is convex if it is possible to take convex combinations of points.