face of a convex set

face of a convex set

Let

C

be a

convex set

in

ℝ

n

(or any

topological vector
space

). A

face

of

C

is a subset

F

of

C

such that

1.

F

is convex, and

2.

given any

line segment

L

⊆

C

, if

ri

⁡

(

L

)

∩

F

≠

∅

, then

L

⊆

F

.

Here,

ri

⁡

(

L

)

denotes the

relative interior

of

L

(open

segment

of

L

).

A

zero-dimensional

face of a convex set

C

is called an

extreme
point

of

C

.

This definition formalizes the notion of a face of a

convex polygon

or
a

convex polytope

and generalizes it to an arbitrary convex set. For
example, any point on the

boundary

of a

closed unit disk

in

ℝ

2

is its face (and an extreme point).

Observe that the

empty set

and

C

itself are faces of

C

. These
faces are sometimes called

improper faces

, while other faces
are called

proper faces

.

Remarks

.
Let

C

be a convex set.

•

The

intersection

of two faces of

C

is a face of

C

.

•

A face of a face of

C

is a face of

C

.

•

Any proper face of

C

lies on its relative boundary,

rbd

⁡

(

C

)

.

•

The set

Part

⁡

(

C

)

of all relative interiors of the faces of

C

partitions

C

.

•

If

C

is

compact

, then

C

is the

convex hull

of its extreme points.

•

The set

F

⁢

(

C

)

of faces of a convex set

C

forms a

lattice

, where the meet is the intersection:

F

1

∧

F

2

:=

F

1

∩

F

2

; the join of

F

1

,

F

2

is the smallest face

F

∈

F

⁢

(

C

)

containing both

F

1

and

F

2

. This lattice is

bounded lattice

(by

∅

and

C

). And it is not hard to see that

F

⁢

(

C

)

is a

complete lattice

.

•

However, in general,

F

⁢

(

C

)

is not a

modular lattice

. As a counterexample, consider the

unit square

[

0

,

1

]

×

[

0

,

1

]

and faces

a

=

(

0

,

0

)

,

b

=

{

(

0

,

y

)

∣

y

∈

[

0

,

1

]

}

, and

c

=

(

1

,

1

)

. We have

a

≤

b

. However,

a

∨

(

b

∧

c

)

=

(

0

,

0

)

∨

∅

=

(

0

,

0

)

, whereas

(

a

∨

b

)

∧

c

=

b

∧

∅

=

∅

.

•

Nevertheless,

F

⁢

(

C

)

is a

complemented lattice

. Pick any face

F

∈

F

⁢

(

C

)

. If

F

=

C

, then

∅

is a

complement

of

F

. Otherwise, form

Part

⁡

(

C

)

and

Part

⁡

(

F

)

, the partitions of

C

and

F

into

disjoint unions

of the relative interiors of their corresponding faces. Clearly

Part

⁡

(

F

)

⊂

Part

⁡

(

C

)

strictly. Now, it is possible to find an extreme point

p

such that

{

p

}

∈

Part

⁡

(

C

)

-

Part

⁡

(

F

)

. Otherwise, all extreme points lie in

Part

⁡

(

F

)

, which leads to

Part

⁡

(

F

)

=

Part

⁡

(

convex hull of extreme points of

⁢

C

)

=

Part

⁡

(

C

)

,

a

contradiction

. Finally, let

G

be the convex hull of extreme points of

C

not

contained

in

Part

⁡

(

F

)

. We assert that

G

is a complement of

F

. If

x

∈

G

∩

F

, then

G

∩

F

is a proper face of

G

and of

F

, hence its extreme points are also extreme points of

G

, and of

F

, which is impossible by the construction of

G

. Therefore

F

∩

G

=

∅

. Next, note that the union of extreme points of

G

and of

F

is the

collection

of all extreme points of

C

, this is again the result of the construction of

G

, so any

y

∈

C

is in the join of all its extreme points, which is equal to the join of

F

and

G

(since join is universally associative).

•

Additionally, in

F

⁢

(

C

)

, zero-dimensional faces are

compact elements

, and compact elements are faces with finitely many extreme points. The

unit disk

D

is not compact in

F

⁢

(

D

)

. Since every face is the convex hull (join) of all extreme points it contains,

F

⁢

(

C

)

is an

algebraic lattice

.

References

1

R.T. Rockafellar,

Convex Analysis

, Princeton University Press, 1996.

Title

face of a convex set

Canonical name

FaceOfAConvexSet

Date of creation

2013-03-22 16:23:08

Last modified on

2013-03-22 16:23:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 52A99

Related topic

ExtremePoint

Defines

face

Defines

proper face

Defines

extreme point

Defines

improper face

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