Editing Convex set (section)

=== Convex hulls ===
{{Main|convex hull}}
Every subset {{mvar|A}} of the vector space is contained within a smallest convex set (called the ''convex hull'' of {{mvar|A}}), namely the intersection of all convex sets containing {{mvar|A}}. The convex-hull operator Conv() has the characteristic properties of a [[closure operator]]:
* ''extensive'': {{math|''S''&nbsp;⊆&nbsp;Conv(''S'')}},
* ''[[Monotone function#Monotonicity in order theory|non-decreasing]]'': {{math|''S''&nbsp;⊆&nbsp;''T''}} implies that {{math|Conv(''S'')&nbsp;⊆&nbsp;Conv(''T'')}}, and
* ''[[idempotence|idempotent]]'': {{math|Conv(Conv(''S'')) {{=}} Conv(''S'')}}.
The convex-hull operation is needed for the set of convex sets to form a <!-- complete  -->[[lattice (order)|lattice]], in which the [[join and meet|"''join''" operation]] is the convex hull of the union of two convex sets
<math display=block>\operatorname{Conv}(S)\vee\operatorname{Conv}(T) = \operatorname{Conv}(S\cup T) = \operatorname{Conv}\bigl(\operatorname{Conv}(S)\cup\operatorname{Conv}(T)\bigr).</math>
The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete [[lattice (order)|lattice]].