aarjaneiro/planetmath huggingface dataset:
exposed points are dense in the extreme points

exposed points are dense in the extreme points

Definition

.

Let

K

⊂

ℝ

n

be a closed

convex set

. A point

p

∈

K

is called an

exposed

point if there is an

n

-

1

dimensional

hyperplane

whose intersection with

K

is

p

alone.

Theorem

(Strasziewicz).

Let

K

⊂

R

n

be a closed convex set. Then the set of

exposed points

is dense in the set
of

extreme points

.

For example, let

C

⁢

(

p

)

denote the closed ball in

ℝ

2

of radius 1 and centered at

p

.

Then take

K

to be the

convex hull

of

C

⁢

(

-

1

,

0

)

and

C

⁢

(

1

,

0

)

. The points

(

-

1

,

1

)

,

(

-

1

,

-

1

)

,

(

1

,

1

)

,

and

(

1

,

-

1

)

are
extreme points, but they are not exposed points.

Title

exposed points are dense in the extreme points

Canonical name

ExposedPointsAreDenseInTheExtremePoints

Date of creation

2013-03-22 17:41:04

Last modified on

2013-03-22 17:41:04

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 52A99

Synonym

Strasziewicz theorem

Related topic

ExtremePoint

Defines

exposed point

Generated on Fri Feb 9 22:16:21 2018 by

LaTeXML
extreme point

extreme point

Definition

.

Let

C

be a

convex subset

of a

vector space

X

. A point

x

∈

C

is
called an

extreme point

if it is not an

interior point

of any

line segment

in

C

. That is

x

is extreme if and only if whenever

x

=

t

⁢

y

+

(

1

-

t

)

⁢

z

,

t

∈

(

0

,

1

)

,

z

≠

y

, implies either

y

∉

C

or

z

∉

C

.

For example the set

[

0

,

1

]

∈

ℝ

is a

convex set

and

0

and

1

are the extreme points.

References

1

H. L. Royden.

. Prentice-Hall, Englewood Cliffs, New Jersey, 1988

Title

extreme point

Canonical name

ExtremePoint

Date of creation

2013-03-22 14:24:55

Last modified on

2013-03-22 14:24:55

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 52A99

Related topic

FaceOfAConvexSet

Related topic

ExposedPointsAreDenseInTheExtremePoints

Generated on Fri Feb 9 22:16:22 2018 by

LaTeXML
extreme subset of convex set

extreme subset of convex set

Let

K

a non-empty closed

convex subset

(

http://planetmath.org/ConvexSet

) of a

normed vector space

. A set

A

⊆

K

is called an

extreme subset

of

K

if

A

is closed, convex and satisfies the condition

:

for any

x

,

y

∈

K

and

t

⁢

x

+

(

1

-

t

)

⁢

y

∈

A

,

t

∈

(

0

,

1

)

then

x

,

y

∈

A

.

For example let

K

=

[

0

,

1

]

×

[

0

,

1

]

then

K

, sides of

K

, included the

endpoints

, and

{

(

1

,

1

)

,

(

0

,

1

)

,

(

1

,

0

)

,

(

0

,

0

)

}

are extreme subsets of

K

.

Title

extreme subset of convex set

Canonical name

ExtremeSubsetOfConvexSet

Date of creation

2013-03-22 15:24:43

Last modified on

2013-03-22 15:24:43

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

7

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 52A99

Related topic

ConvexSet

Generated on Fri Feb 9 22:16:24 2018 by

LaTeXML
face of a convex set, alternative definition of

face of a convex set, alternative definition of

The following definition of a face of a

convex set

in a

real vector
space

is sometimes useful.

Let

C

be a

convex subset

of

ℝ

n

. Before we define faces,
we introduce oriented

hyperplanes

and

supporting hyperplanes

.

Given any vectors

n

and

p

in

ℝ

n

, define the hyperplane

H

⁢

(

n

,

p

)

by

H

⁢

(

n

,

p

)

=

{

x

∈

ℝ

n

:

n

⋅

(

x

-

p

)

=

0

}

;

note that this is the degenerate hyperplane

ℝ

n

if

n

=

0

.
As long as

H

⁢

(

n

,

p

)

is

nondegenerate

, its removal disconnects

ℝ

n

. The

upper halfspace

of

ℝ

n

determined by

H

⁢

(

n

,

p

)

is

H

⁢

(

n

,

p

)

+

=

{

x

∈

ℝ

n

:

n

⋅

(

x

-

p

)

≥

0

}

.

A hyperplane

H

⁢

(

n

,

p

)

is a

supporting hyperplane

for

C

if its upper halfspace contains

C

, that is, if

C

⊂

H

(

n

.

p

)

+

.

Using this terminology, we can define a

face

of a convex set

C

to be the intersection of

C

with a supporting hyperplane of

C

.
Notice that we still get the empty set and

C

as

improper faces

of

C

.

Remarks.

Let

C

be a convex set.

•

If

F

1

=

C

∩

H

⁢

(

n

1

,

p

1

)

and

F

2

=

C

∩

H

⁢

(

n

2

,

p

2

)

are faces
of

C

intersecting in a point

p

, then

H

⁢

(

n

1

+

n

2

,

p

)

is a
supporting hyperplane of

C

, and

F

1

∩

F

2

=

C

∩

H

⁢

(

n

1

+

n

2

,

p

)

.
This shows that the faces of

C

form a meet-semilattice.

•

Since each proper face lies on the base of the upper halfspace of some
supporting hyperplane, each such face must lie on the

relative
boundary

of

C

.

Title

face of a convex set, alternative definition of

Canonical name

FaceOfAConvexSetAlternativeDefinitionOf

Date of creation

2013-03-22 17:02:02

Last modified on

2013-03-22 17:02:02

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Definition

Classification

msc 52A99

Synonym

face

Defines

supporting hyperplane

Generated on Fri Feb 9 22:16:37 2018 by

LaTeXML
if A is convex and f linear then f⁢(A) and f^{-1}⁢(A) are convex

if

A

is convex and

f

linear then

f

⁢

(

A

)

and

f

-

1

⁢

(

A

)

are convex

Proposition 1

.

Suppose

X

,

Y

are

vector spaces

over

R

(or

C

),
and suppose

f

:

X

→

Y

is a

linear map

.

1.

If

A

⊆

X

is convex, then

f

⁢

(

A

)

is convex.

2.

If

B

⊆

Y

is convex, then

f

-

1

⁢

(

B

)

is convex,
where

f

-

1

is the

inverse image

.

Proof.

For the first claim,
suppose

y

,

y

′

∈

f

⁢

(

A

)

, say,

y

=

f

⁢

(

x

)

and

y

′

=

f

⁢

(

x

′

)

for

x

,

x

′

∈

A

, and suppose

λ

∈

(

0

,

1

)

.
Then

λ

⁢

y

+

(

1

-

λ

)

⁢

y

′

=

λ

⁢

f

⁢

(

x

)

+

(

1

-

λ

)

⁢

f

⁢

(

x

′

)

=

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

x

′

)

,

so

λ

⁢

y

+

(

1

-

λ

)

⁢

y

′

∈

f

⁢

(

A

)

as

A

is convex.

For the second claim, let us first recall that

x

∈

f

-

1

⁢

(

B

)

if and only if

f

⁢

(

x

)

∈

B

. Then, if

x

,

x

′

∈

f

-

1

⁢

(

B

)

,
and

λ

∈

(

0

,

1

)

, we have

f

⁢

(

λ

⁢

x

+

(

1

-

λ

)

⁢

x

′

)

=

λ

⁢

f

⁢

(

x

)

+

(

1

-

λ

)

⁢

f

⁢

(

x

′

)

.

As

B

is convex, the

right hand side

belongs to

B

,
and

λ

⁢

x

+

(

1

-

λ

)

⁢

x

′

∈

f

-

1

⁢

(

B

)

.
∎

Title

if

A

is convex and

f

linear then

f

⁢

(

A

)

and

f

-

1

⁢

(

A

)

are convex

Canonical name

IfAIsConvexAndFLinearThenFAAndF1AAreConvex

Date of creation

2013-03-22 14:36:18

Last modified on

2013-03-22 14:36:18

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Theorem

Classification

msc 52A99

Related topic

InverseImage

Related topic

DirectImage

Generated on Fri Feb 9 22:16:40 2018 by

LaTeXML
internal point

internal point

Definition

.

Let

X

be a

vector space

and

S

⊂

X

. Then

x

∈

S

is called an

internal point

of

S

if and only if the

intersection

of each line in

X

through

x

and

S

contains a small

interval

around

x

.

That is

x

is an internal point of

S

if whenever

y

∈

X

there exists an

ϵ

>

0

such that

x

+

t

⁢

y

∈

S

for all

t

<

ϵ

.

If

X

is a

topological vector space

and

x

is in the

interior

of

S

, then it is an internal point, but the

converse

is not true in general. However if

S

⊂

ℝ

n

is a

convex set

then all internal points are

interior points

and vice versa.

References

1

H. L. Royden.

. Prentice-Hall, Englewood Cliffs, New Jersey, 1988

Title

internal point

Canonical name

InternalPoint

Date of creation

2013-03-22 14:25:04

Last modified on

2013-03-22 14:25:04

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 52A99

Generated on Fri Feb 9 22:16:41 2018 by

LaTeXML
proof of properties of extreme subsets of a closed convex set

proof of properties of extreme subsets of a closed convex set

For the first claim, it is obvious that

⋂

i

∈

I

A

i

is closed

convex subset

(

http://planetmath.org/ConvexSet

) of

K

. Let

z

∈

A

and

0

<

t

<

1

,

x

,

y

∈

K

such as

z

=

t

⁢

x

+

(

1

-

t

)

⁢

y

. Then

z

∈

A

i

, for all

i

∈

I

so we have that

x

,

y

∈

A

i

for all

i

∈

I

. Therefore

x

,

y

∈

⋂

i

∈

I

A

i

.

For the second claim suppose

x

,

y

∈

K

,

t

∈

(

0

,

1

)

and

z

∈

A

such as

z

=

t

⁢

x

+

(

1

-

t

)

⁢

y

. From the hypothesis

A

⊂

B

we have that

z

∈

B

and since

B

is an extreme subset of

K

,

x

,

y

∈

B

. Analogously from the hypothesis that

A

is an extreme subset of

B

, we have that

x

,

y

∈

A

.

Title

proof of properties of extreme subsets of a closed

convex set

Canonical name

ProofOfPropertiesOfExtremeSubsetsOfAClosedConvexSet

Date of creation

2013-03-22 15:25:12

Last modified on

2013-03-22 15:25:12

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

4

Author

georgiosl (7242)

Entry type

Proof

Classification

msc 52A99

Generated on Fri Feb 9 22:16:43 2018 by

LaTeXML
properties of extreme subsets of a closed convex set

properties of extreme subsets of a closed convex set

Let

K

be a closed

convex subset

(

http://planetmath.org/ConvexSet

) of a

normed vector space

1.

If

{

A

i

:

i

∈

I

}

is a family of extreme subsets of

K

, such as

⋂

i

∈

I

A

i

≠

∅

then

⋂

i

∈

I

A

i

is extreme subset of

K

2.

A

⊂

B

⊂

K

such as

A

,

B

are extreme subsets of

B

and

K

respectively. Then

A

is an extreme subset of

K

.

Title

properties of extreme subsets of a closed

convex set

Canonical name

PropertiesOfExtremeSubsetsOfAClosedConvexSet

Date of creation

2013-03-22 15:24:46

Last modified on

2013-03-22 15:24:46

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

7

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 52A99

Related topic

ConvexSet

Generated on Fri Feb 9 22:16:45 2018 by

LaTeXML
isothetic

isothetic

A curve is

isothetic

if it consists entirely of lines

parallel

to one of the

coordinate

axes in a given rectilinear

coordinate system

. A

polygon

or

polyhedron

is isothetic if all of its edges are parallel to one of the coordinate axes.

An example of an isothetic polygon is the

rectangle

{

(

x

,

y

)

:

x

1

≤

x

≤

x

2

,

y

1

≤

y

≤

y

2

}

for some

x

1

,

x

2

,

y

1

,

y

2

. Examples of non-isothetic shapes are the tilted square

{

(

x

,

y

,

z

)

:

|

x

|

+

|

y

|

=

1

}

and the bipyramid

{

(

x

,

y

,

z

)

:

|

x

|

+

|

y

|

+

|

z

|

=

1

}

.

(This entry is here because I couldn’t find a definition of isothetic on the web. If you know anything interesting about isothetic shapes, please adopt this entry!)

Title

isothetic

Canonical name

Isothetic

Date of creation

2013-03-22 15:54:33

Last modified on

2013-03-22 15:54:33

Owner

lha (3057)

Last modified by

lha (3057)

Numerical id

8

Author

lha (3057)

Entry type

Definition

Classification

msc 52B11

Generated on Fri Feb 9 22:16:47 2018 by

LaTeXML
f-vector

f-vector

Let

P

be a

polytope

of

dimension

d

. The

f-vector

of

P

is the finite

integer sequence

(

f

0

,

…

,

f

d

-

i

)

, where
the

component

in position

i

is the number of

i

-dimensional
faces of

P

. For some purposes it is convenient to view the empty
face and the polytope itself as

improper faces

, so

f

-

1

=

f

d

=

1

.

For example, a cube has 8 vertices, 12 edges, and 6 faces, so its
f-vector is (8, 12, 6).

The entries in the f-vector of a convex polytope satisfy the
Euler–Poincaré–Schläfli formula:

∑

-

1

≤

i

≤

d

(

-

1

)

i

⁢

f

i

=

0

.

Consequently, the face

lattice

of a polytope is Eulerian. For any

graded poset

with maximum and minimum elements there is an

extension

of the f-vector called the

flag f-vector

. For any subset

S

of

{

0

,

1

,

…

,

d

-

1

}

, the

f

S

entry of the flag
f-vector of

P

is the number of chains of faces in

ℒ

⁢

(

P

)

with dimensions coming only from

S

.

The flag f-vector of a three-dimensional cube is given in the
following table. For simplicity we drop braces and commas.

S

f

S

∅

1

0

8

1

12

2

6

01

8

⋅

3

=

24

02

8

⋅

3

=

24

12

12

⋅

2

=

24

012

8

⋅

3

⋅

2

=

48

For example,

f

{

1

,

2

}

=

24

because each of the 12 edges
meets exactly two faces.

Although the flag f-vector of a

d

-polytope has

2

d

entries,
most of them are

redundant

, as they satisfy a collection of

identities

generalizing the Euler–Poincaré–Schläfli formula and called the
generalized Dehn-Sommerville

relations

. Interestingly, the number of
nonredundant entries in the flag

f

-vector of a

d

-polytope is
one less than the

Fibonacci number

F

d

-

1

.

References

1

Bayer, M. and L. Billera,

Generalized Dehn-Sommerville relations for
polytopes, spheres and Eulerian

partially ordered sets

, Invent. Math. 79
(1985), no. 1, 143–157.

2

Bayer, M. and A. Klapper,

A new index for polytopes

, Discrete Comput.
Geom. 6
(1991), no. 1, 33–47.

3

Ziegler, G.,

Lectures on polytopes

, Springer-Verlag, 1997.

Title

f-vector

Canonical name

Fvector

Date of creation

2013-03-22 16:59:10

Last modified on

2013-03-22 16:59:10

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Definition

Classification

msc 52B40

Synonym

f

-vector

Defines

flag f-vector

Defines

flag

f

-vector

Generated on Fri Feb 9 22:16:50 2018 by

LaTeXML
polytope

polytope

A

polytope

is the

convex hull

of finitely many points in

Euclidean space

. A polytope constructed in this way is the convex
hull of its vertices and is called a

V

-polytope

.
An

H

-polytope

is a

bounded

intersection

of
upper halfspaces. By the Weyl–Minkowski theorem, these descriptions are

equivalent

, that is, every

𝒱

-polytope is an

ℋ

-polytope, and vice versa. This shows that our
intuition, based on the study of low-dimensional polytopes, that one
can describe a polytope either by its vertices or by its facets is
essentially correct.

The

dimension

of

P

is the smallest

d

such that

P

can be
embedded in

ℝ

d

. A

d

-dimensional polytope is also called
a

d

-polytope

.

A face of a polytope is the intersection of the polytope with a

supporting hyperplane

. Intuitively, a supporting hyperplane is a

hyperplane

that “just touches” the polytope, as though the polytope
were just about to

pass through

the hyperplane. Note that this
intuitive picture does not cover the case of the empty face, where the
supporting hyperplane does not touch the polytope at all, or the fact
that a polytope is a face of itself. The faces of a polytope, when
partially ordered by

set inclusion

, form a

geometric lattice

, called
the face

lattice

of the polytope.

The

Euler polyhedron

formula

, which states that if a

3

-polytope has

V

vertices,

E

edges, and

F

faces, then

V

-

E

+

F

=

2

,

has a

generalization

to all

d

-polytopes. Let

(

f

-

1

=

1

,

f

0

,

…

,

f

d

-

1

,

f

d

=

1

)

be the

f-vector

of a

d

-polytope

P

, so

f

i

is the number of

i

-dimensional faces of

P

. Then these numbers
satsify the Euler–Poincaré–Schläfli formula:

∑

i

=

-

1

d

(

-

1

)

i

⁢

f

i

=

0

.

(1)

This is the first of many

relations

among entries of the f-vector
satisfied by all polytopes. These relations are called the

Dehn–Sommerville relations

. Any poset which

satisfies

these
relations is Eulerian (

http://planetmath.org/EulerianPoset

),
so the face lattice of any polytope is Eulerian.

References

1

Bayer, M. and L. Billera,

Generalized Dehn–Sommerville relations for
polytopes, spheres and Eulerian

partially ordered sets

, Invent. Math. 79
(1985), no. 1, 143–157.

2

Bayer, M. and A. Klapper,

A new index for polytopes

, Discrete Comput.
Geom. 6
(1991), no. 1, 33–47.

3

Minkowski, H.

Allgemeine Lehrsätze über die konvexe Polyeder

, Nachr. Ges. Wiss., Göttingen, 1897, 198–219.

4

Weyl, H.

Elementare Theorie der konvexen Polyeder

, Comment. Math. Helvetici, 1935, 7

5

Ziegler, G.,

Lectures on polytopes

, Springer-Verlag, 1997.

Title

polytope

Canonical name

Polytope

Date of creation

2013-03-22 14:07:59

Last modified on

2013-03-22 14:07:59

Owner

mps (409)

Last modified by

mps (409)

Numerical id

26

Author

mps (409)

Entry type

Definition

Classification

msc 52B40

Related topic

Polyhedron

Related topic

PoincareFormula

Related topic

EulersPolyhedronTheorem

Defines

V-polytope

Defines

H-polytope

Defines

d-polytope

Defines

dimension

Generated on Fri Feb 9 22:16:52 2018 by

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Weyl-Minkowski theorem

Weyl-Minkowski theorem

Theorem

(Weyl–Minkowski).

A subset of

Euclidean space

is a

convex polytope

if and only if it is a

bounded polyhedron

.

References

1

Minkowski, H.

Allgemeine Lehrsätze über die konvexe Polyeder

, Nachr. Ges. Wiss., Göttingen, 1897, 198–219.

2

Weyl, H.

Elementare Theorie der konvexen Polyeder

, Comment. Math. Helvetici, 1935, 7

3

Ziegler, G.,

Lectures on

polytopes

, Springer-Verlag, 1997.

Title

Weyl-Minkowski theorem

Canonical name

WeylMinkowskiTheorem

Date of creation

2013-03-22 16:59:07

Last modified on

2013-03-22 16:59:07

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Theorem

Classification

msc 52B40

Synonym

Weyl–

Minkowski theorem

Generated on Fri Feb 9 22:16:54 2018 by

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hyperplane arrangement

hyperplane arrangement

Let

V

be a

vector space

over a field

𝕂

. A

hyperplane

arrangment

in

V

is a family

𝒜

=

{

ℋ

i

}

i

∈

I

of affine hyperplanes in

V

. If all of the hyperplanes
pass through

0

,

𝒜

is called

central

;
otherwise, it is

affine

. More generally, a

subspace arrangement

is a family of affine subspaces of

V

.
The same distinction between central and affine subspace arrangement
holds.

Example 1

.

Let

V

=

K

n

. Then the family

𝕂

⁢

P

n

=

{

S

⊂

V

∣

dim

𝕂

⁡

(

S

)

=

1

}

of

1

-dimensional subspaces of

V

is a central subspace
arrangement, the

projective space

of

dimension

n

over

K

.

Instead of considering all lines through a vector space,
we could consider all

k

-dimensional subspaces
of the space.

Example 2

.

Again let

V

=

K

n

, and suppose

0

≤

k

≤

n

. Then the
family

Gr

⁡

(

V

,

k

)

=

{

S

⊂

V

∣

dim

𝕂

⁡

(

S

)

=

k

}

of

k

-dimensional subspaces of

V

is a central subspace
arrangement, the

Grassmannian

. Observe that

K

⁢

P

n

=

Gr

⁡

(

K

n

,

1

)

.

If

V

is a

topological vector space

and

𝒜

is a
hyperplane arrangement, then it makes sense to ask for the

fundamental group

of the

complement

V

∖

⋃

ℋ

∈

𝒜

ℋ

.

Example 3

.

If

A

is a finite hyperplane arrangement over

V

=

R

n

, then the arrangement

partitions

(

http://planetmath.org/Partition

)

V

into a finite number of

contractible

cells. By selecting
a point in each cell and taking the convex hull of the result,
we obtain a polytope combinatorially

equivalent

to the
zonotope dual to the arrangement. Since the question of
the fundamental group here is not interesting, we could also
use the

embedding

R

n

↪

C

n

to complexify

A

. In this case the
complement

C

n

∖

⋃

H

∈

A

H

usually has nontrivial fundamental group.

References

1

Klain, D. A., and G.-C. Rota,

,

Introduction

to geometric probability

,
Cambridge University Press, 1997.

2

Orlik, P., and H. Terao,

Arrangements of hyperplanes

,
Springer-Verlag, 1992.

Title

hyperplane arrangement

Canonical

name

HyperplaneArrangement

Date of creation

2013-03-22 15:47:55

Last modified on

2013-03-22 15:47:55

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Definition

Classification

msc 52B99

Classification

msc 52C35

Synonym

subspace arrangement

Synonym

central arrangement

Related topic

Zonotope

Defines

Grassmannian

Generated on Fri Feb 9 22:16:56 2018 by

LaTeXML
intersection semilattice of a subspace arrangement

intersection semilattice of a subspace arrangement

Let

𝒜

be a finite

subspace arrangement

in a

finite-dimensional

vector space

V

.
The of

𝒜

is the subspace
arrangement

L

⁢

(

𝒜

)

defined by taking the

closure

(

http://planetmath.org/ClosureAxioms

)
of

𝒜

under

intersections

. More formally, let

L

⁢

(

𝒜

)

=

{

⋂

H

∈

𝒮

H

∣

𝒮

⊂

𝒜

}

.

Order (

http://planetmath.org/Poset

) the

elements of

L

⁢

(

𝒜

)

by reverse inclusion,
and give it the structure of a

join-semilattice

by defining

H

∨

K

=

H

∩

K

for all

H

,

K

in

L

⁢

(

𝒜

)

.
Moreover, the elements of

L

⁢

(

𝒜

)

are naturally
graded by codimension. If

𝒜

happens to be a central arrangement, its

intersection
semilattice

is in fact a

lattice

, with the meet operation
defined by

H

∧

K

=

span

⁡

(

H

∪

K

)

, where

span

⁡

(

H

∪

K

)

is the

subspace

of

V

spanned by

H

∪

K

.

Title

intersection semilattice of a subspace arrangement

Canonical name

IntersectionSemilatticeOfASubspaceArrangement

Date of creation

2013-03-22 15:47:58

Last modified on

2013-03-22 15:47:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 52B99

Classification

msc 52C35

Synonym

intersection lattice

Synonym

intersection semilattice

Generated on Fri Feb 9 22:16:57 2018 by

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zonotope

zonotope

A

zonotope

is a

polytope

which can be obtained as the

Minkowski sum

(

http://planetmath.org/MinkowskiSum3

) of finitely many

closed line segments

in

ℝ

n

. Three-dimensional zonotopes are also sometimes called

zonohedra

. Zonotopes are dual to finite

hyperplane arrangements

. They are centrally

symmetric

, compact,

convex sets

.

For example, the unit

n

-cube is the Minkowski sum of
the

line segments

from the origin to the standard unit vectors

e

i

for

1

≤

i

≤

n

.
A

hexagon

is also a zonotope; for example, the Minkowski
sum of the line segments based at the origin with

endpoints

at

(

1

,

0

)

,

(

1

,

1

)

, and

(

0

,

1

)

is a hexagon. In fact, any projection of an

n

-cube is a zonotope.

The prism of a zonotope is always a zonotope, but the

pyramid

of a
zonotope need not be. In particular, the

n

-

simplex

(

http://planetmath.org/HomologyTopologicalSpace

) is only a
zonotope for

n

≤

1

.

References

1

Billera, L., R. Ehrenborg, and M. Readdy,

The

cd

-index of zonotopes and

arrangements

, in

Mathematical essays in honor of Gian-Carlo Rota

, (B. E. Sagan and R. P. Stanley, eds.), BirkhÃÂ¤user, Boston, 1998, pp. 23–40.

2

Ziegler, G.,

Lectures on polytopes

, Springer-Verlag, 1997.

Title

zonotope

Canonical name

Zonotope

Date of creation

2013-03-22 15:47:20

Last modified on

2013-03-22 15:47:20

Owner

mps (409)

Last modified by

mps (409)

Numerical id

7

Author

mps (409)

Entry type

Definition

Classification

msc 52B99

Synonym

zonohedron

Synonym

zonohedra

Related topic

HyperplaneArrangement

Generated on Fri Feb 9 22:16:59 2018 by

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