Convex set: Difference between revisions

Revision as of 19:32, 15 April 2023

Revision as of 19:38, 15 April 2023

Line 14:

== Definitions ==

[[File:Convex supergraph.svg|right|thumb|A [[convex function|function]] is convex if and only if its [[Epigraph (mathematics)|epigraph]], the region (in green) above its [[graph of a function|graph]] (in blue), is a convex set.]]

Let {{mvar|S}} be a [[vector space]] or an [[affine space]] over the [[real number]]s, or, more generally, over some [[ordered field]] (this includes Euclidean spaces, which are affine spaces). A [[subset]] {{mvar|C}} of {{mvar|S}} is '''convex''' if, for all {{mvar|x}} and {{mvar|y}} in {{mvar|C}}, the [[line segment]] connecting {{mvar|x}} and {{mvar|y}} is included in {{mvar|C}}.

This means that the [[affine combination]] {{math|(1 − ''t'')''x'' + ''ty''}} belongs to {{mvar|C}}, for all {{mvar|x}} and {{mvar|,y}} in {{mvar|C}}, and {{mvar|t}} in the [[interval (mathematics)|interval]] {{math|[0, 1]}}. This implies that convexity (the property of being convex) is invariant under [[affine transformation]]s. ThisFurther, impliesit alsoimplies that a convex set in a [[real number|real]] or [[complex number|complex]] [[topological vector space]] is [[path-connected]], thus [[connected space|connected]].

A set {{mvar|C}} is ''{{visible anchor|strictly convex}}'' if every point on the line segment connecting {{mvar|x}} and {{mvar|y}} other than the endpoints is inside the [[Interior (topology)|topological interior]] of {{mvar|C}}. A closed convex subset is strictly convex if and only if every one of its [[Boundary (topology)|boundary points]] is an [[extreme point]].<ref>{{Halmos A Hilbert Space Problem Book 1982|p=5}}</ref>