Convex set: Difference between revisions

Revision as of 11:36, 5 February 2025

Revision as of 22:52, 7 February 2025

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belongs to {{mvar|S}}. As the definition of a convex set is the case {{math|1=''r'' = 2}}, this property characterizes convex sets.

Such an affine combination is called a [[convex combination]] of {{math|''u''₁, ..., ''u_r''}}. The '''convex hull''' of a subset {{mvar|S}} of a real vector space is defined as the intersection of all convex sets that contain {{mvar|S}}. More concretely, the convex hull is the set of all convex combinations of points in {{mvar|S}}. In particular, this is a convex set.

A ''(bounded) [[convex polytope]]'' is the convex hull of a finite subset of some Euclidean space {{math|'''R'''^''n''}}.

=== Intersections and unions ===

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A '''face''' of a convex set <math>C</math> is a subset <math>F</math> of <math>C</math> such that <math>F</math> is also a convex set, and for any points <math>x,y</math> in <math>C</math> and any real number <math>0<t<1</math> with <math>(1-t)x+ty</math> in <math>F</math>, <math>x</math> and <math>y</math> must both be in <math>F</math>.{{sfn | Rockafellar| 1997 | p=162}} For example, <math>C</math> itself and the empty set are faces of <math>C</math>; these are sometimes called the ''trivial faces'' of <math>C</math>. An '''[[extreme point]]''' of <math>C</math> is a point that is a face of <math>C</math>.

Let <math>C</math> be a convex set in <math>\R^n</math> that is [[compact space|compact]] (or equivalently, closed and [[bounded set|bounded]]). Then <math>C</math> is the [[convex hull]] of its extreme points.{{sfn | Rockafellar| 1997 | p=166}} This holds more generally for a compact convex set in any [[locally convex topological vector space]] (the [[Krein–Milman theorem]]).

For example:

For example, a [[triangle]] in the plane (including the region inside) is a compact convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extremal points are the three vertices.) For another example: the only nontrivial faces of the [[closed unit disk]] <math>\{ (x,y) \in \R^2: x^2+y^2 \leq 1 \}</math> are its extremal points, namely the points on the [[unit circle]] <math>S^1 = \{ (x,y) \in \R^2: x^2+y^2=1 \}</math>.

* A [[triangle]] in the plane (including the region inside) is a compact convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extreme points are the three vertices.)

For* example, a [[triangle]] in the plane (including the region inside) is a compact convex set. Its nontrivial faces are the three vertices and the three edges. (So the only extremal points are the three vertices.) For another example: theThe only nontrivial faces of the [[closed unit disk]] <math>\{ (x,y) \in \R^2: x^2+y^2 \leq 1 \}</math> are its extremalextreme points, namely the points on the [[unit circle]] <math>S^1 = \{ (x,y) \in \R^2: x^2+y^2=1 \}</math>.

=== Convex sets and rectangles ===

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=== 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|hull 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

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* [[Complex convexity]]

* [[Convex cone]]

* [[Convex hull]]

* [[Convex series]]

* [[Convex metric space]]