cone

cone

Definition 1

.

Suppose

V

is a real (or complex)

vector space

with a subset

C

.

1.

If

λ

⁢

C

⊂

C

for any real

λ

>

0

,
then

C

is called a

cone

.

2.

If the origin belongs to a cone, then the cone is said to be

pointed

.
Otherwise, the cone is

blunt

.

3.

A pointed cone is

salient

, if it contains no

1

-dimensional

vector subspace

of

V

.

4.

If

C

-

x

0

is a cone for some

x

0

in

V

,
then

C

is a

cone with vertex

at

x

0

.

5.

A convex pointed cone is called a

wedge

.

6.

A

proper cone

is a convex cone

C

with vertex at

0

, such that

C

∩

(

-

C

)

=

{

0

}

. A slightly more specific definition of a proper cone is this entry (

http://planetmath.org/ProperCone

), but it requires the vector space to be topological.

7.

A cone

C

is said to be

generating

if

V

=

C

-

C

. In this case,

V

is said to be

generated by

C

.

Examples

1.

In

ℝ

, the set

x

>

0

is a blunt cone.

2.

In

ℝ

, the set

x

≥

0

is a pointed salient cone.

3.

Suppose

x

∈

ℝ

n

. Then for any

ε

>

0

, the set

C

=

⋃

{

λ

⁢

B

x

⁢

(

ε

)

⁢

∣

λ

>

⁢

0

}

is an open cone. If

|

x

|

<

ε

, then

C

=

ℝ

n

.
Here,

B

x

⁢

(

ε

)

is the open ball at

x

with radius

ε

.

4.

In a

normed vector space

, a blunt cone

C

is completely
determined by the

intersection

of

C

with the unit sphere.

Properties

1.

The union and intersection of a

collection

of cones is a cone. In other words, the set of cones forms a

complete lattice

.

2.

The

complement

of a cone is a cone. This means that the complete lattice of cones is also a

complemented lattice

.

3.

A cone

C

is convex iff

C

+

C

⊆

C

.

Proof.

If

C

is convex and

a

,

b

∈

C

, then

1

2

⁢

a

,

1

2

⁢

b

∈

C

, so their sum, being the

convex combination

of

a

,

b

, is in

C

, and therefore

a

+

b

=

2

⁢

(

1

2

⁢

a

+

1

2

⁢

b

)

∈

C

also.

Conversely

, suppose a cone

C

satisfies

C

+

C

⊆

C

, and

a

,

b

∈

C

. Then

λ

⁢

a

,

(

1

-

λ

)

⁢

b

∈

C

for

λ

>

0

(the case when

λ

=

0

is obvious). Therefore their sum is also in

C

.
∎

4.

A cone containing

0

is a cone with vertex at

0

. As a result, a wedge is a cone with vertex at

0

.

5.

The only cones that are

subspaces

at the same time are wedges.

References

1

M. Reed, B. Simon,

Methods of Modern Mathematical Physics:

Functional Analysis

I

,
Revised and enlarged edition, Academic Press, 1980.

2

J. Horváth,

Topological Vector Spaces

and

Distributions

,
Addison-Wesley Publishing Company, 1966.

3

R.E. Edwards,

Functional Analysis: Theory and Applications

,
Dover Publications, 1995.

4

I.M. Glazman, Ju.I. Ljubic,

Finite-Dimensional

Linear

Analysis

, A systematic

Presentation

in Problem Form

,
Dover Publications, 2006.

Title

cone

Canonical name

Cone1

Date of creation

2013-03-22 15:32:58

Last modified on

2013-03-22 15:32:58

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

16

Author

matte (1858)

Entry type

Definition

Classification

msc 46-00

Related topic

ProperCone

Related topic

GeneralizedFarkasLemma

Defines

blunt cone

Defines

pointed cone

Defines

salient cone

Defines

cone with vertex

Defines

wedge

Defines

proper cone

Defines

generating

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