Convex set: Difference between revisions

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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. This implies also 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]].{{sfn|Halmos|1982|p=5}}

A set {{mvar|C}} is ''[[absolutely convex]]'' if it is convex and [[balanced set|balanced]].