aarjaneiro/planetmath huggingface dataset:
convex combination

convex combination

Let

V

be some

vector space

over

ℝ

. Let

X

be some set of elements of

V

. Then a

convex combination

of elements from

X

is a

linear combination

of the form

λ

1

⁢

x

1

+

λ

2

⁢

x

2

+

⋯

+

λ

n

⁢

x

n

for some

n

>

0

, where each

x

i

∈

X

, each

λ

i

≥

0

and

∑

i

λ

i

=

1

.

Let

co

⁢

(

X

)

be the set of all convex combinations from

X

. We call

co

⁢

(

X

)

the

convex hull

, or

convex envelope

, or

convex closure

of

X

. It is a

convex set

, and is the smallest convex set which contains

X

. A set

X

is convex if and only if

X

=

co

⁢

(

X

)

.

Title

convex combination

Canonical name

ConvexCombination

Date of creation

2013-03-22 11:50:36

Last modified on

2013-03-22 11:50:36

Owner

mps (409)

Last modified by

mps (409)

Numerical id

14

Author

mps (409)

Entry type

Definition

Classification

msc 52A01

Synonym

convex hull

Synonym

convex envelope

Synonym

convex closure

Related topic

ConvexSet

Related topic

AffineCombination

Generated on Fri Feb 9 22:15:27 2018 by

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relative interior

relative interior

Let

S

be a subset of the

n

-dimensional

Euclidean space

ℝ

n

. The

relative interior

of

S

is the interior of

S

considered as a subset of its

affine hull

Aff

⁡

(

S

)

, and is denoted by

ri

⁡

(

S

)

.

The

difference

between the interior and the relative interior of

S

can be illustrated in the following two examples. Consider the closed

unit square

I

2

:=

{

(

x

,

y

,

0

)

∣

0

≤

x

,

y

≤

1

}

in

ℝ

3

. Its interior is

∅

, the

empty set

. However, its relative interior is

ri

⁡

(

I

2

)

=

{

(

x

,

y

,

0

)

∣

0

<

x

,

y

<

1

}

,

since

Aff

⁡

(

I

2

)

is the

x

-

y

plane

{

(

x

,

y

,

0

)

∣

x

,

y

∈

ℝ

}

. Next, consider the closed

unit cube

I

3

:=

{

(

x

,

y

,

z

)

∣

0

≤

x

,

y

,

z

≤

1

}

in

ℝ

3

. The interior and the relative interior of

I

3

are the same:

int

⁡

(

I

3

)

=

ri

⁡

(

I

3

)

=

{

(

x

,

y

,

z

)

∣

0

<

x

,

y

,

z

<

1

}

.

Remarks

.

•

As another example, the relative interior of a point is the point, whereas the interior of a point is

∅

.

•

It is true that if

T

⊆

S

, then

int

⁡

(

T

)

⊆

int

⁡

(

S

)

. However, this is not the case for the relative interior

operator

ri

, as shown by the above two examples:

∅

≠

I

2

⊆

I

3

, but

ri

⁡

(

I

2

)

∩

ri

⁡

(

I

3

)

=

∅

.

•

The companion

concept

of the relative interior of a set

S

is the

relative boundary

of

S

: it is the boundary of

S

in

Aff

⁡

(

S

)

, denoted by

rbd

⁡

(

S

)

. Equivalently,

rbd

⁡

(

S

)

=

S

¯

-

ri

⁡

(

S

)

, where

S

¯

is the

closure

of

S

.

•

S

is said to be

relatively open

if

S

=

ri

⁡

(

S

)

.

•

All of the

definitions

above can be generalized to convex sets in a

topological vector space

.

Title

relative interior

Canonical

name

RelativeInterior

Date of creation

2013-03-22 16:20:07

Last modified on

2013-03-22 16:20:07

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 52A07

Classification

msc 52A15

Classification

msc 51N10

Classification

msc 52A20

Defines

relative boundary

Defines

relatively open

Generated on Fri Feb 9 22:15:40 2018 by

LaTeXML
Carathéodory’s theorem

Carathéodory’s theorem

Suppose a point

p

lies in the

convex hull

of a set

P

⊂

ℝ

d

. Then there is a subset

P

′

⊂

P

consisting of no more than

d

+

1

points such that

p

lies in the convex hull of

P

′

.

For example, if a point

p

is contained in a convex hull of a set

P

⊂

ℝ

2

, then there are three points in

P

that determine the triangle containing

p

, provided, of course, that

P

contains at least three points.

Title

Carathéodory’s theorem

Canonical name

CaratheodorysTheorem

Date of creation

2013-03-22 13:57:43

Last modified on

2013-03-22 13:57:43

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Theorem

Classification

msc 52A20

Related topic

ConvexSet

Generated on Fri Feb 9 22:15:42 2018 by

LaTeXML
proof of Carathéodory’s theorem

proof of Carathéodory’s theorem

The convex hull of

P

consists precisely of the points that can be written as

convex combination

of finitely many number points in

P

. Suppose that

p

is a convex combination of

n

points in

P

, for some integer

n

,

p

=

α

1

⁢

x

1

+

α

2

⁢

x

2

+

…

+

α

n

⁢

x

n

where

α

1

+

…

+

α

n

=

1

and

x

1

,

…

,

x

n

∈

P

. If

n

≤

d

+

1

, then it is already in the required form.

If

n

>

d

+

1

, the

n

-

1

points

x

2

-

x

1

,

x

3

-

x

1

,

…

,

x

n

-

x

1

are

linearly dependent

. Let

β

i

,

i

=

2

,

3

,

…

,

n

, be

real numbers

, which are not all zero, such that

∑

i

=

2

n

β

i

⁢

(

x

i

-

x

1

)

=

0

.

So, there are

n

constants

γ

1

,

…

⁢

γ

n

, not all equal to zero, such that

∑

i

=

1

n

γ

i

⁢

x

i

=

0

,

and

∑

i

=

1

n

γ

i

=

0

.

Let

ℐ

be a subset of indices defined as

{

i

∈

{

1

,

2

,

…

,

n

}

:

γ

i

>

0

}

.

Since

∑

i

=

1

n

γ

i

=

0

, the subset

ℐ

is not empty. Define

a

=

max

i

∈

I

⁡

α

i

/

γ

i

.

Then we have

p

=

∑

i

=

1

n

(

α

i

-

a

⁢

γ

i

)

⁢

x

i

,

which is a convex combination with at least one zero

coefficient

. Therefore, we can assume that

p

can be written as a convex combination of

n

-

1

points in

P

, whenever

n

>

d

+

1

.

After repeating the above process several times, we can express

p

as a convex combination of at most

d

+

1

points in

P

.

Title

proof of Carathéodory’s theorem

Canonical name

ProofOfCaratheodorysTheorem

Date of creation

2013-03-22 17:50:08

Last modified on

2013-03-22 17:50:08

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

4

Author

kshum (5987)

Entry type

Proof

Classification

msc 52A20

Generated on Fri Feb 9 22:15:44 2018 by

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proper cone

proper cone

A

proper cone

is a cone (

http://planetmath.org/Cone3

)

C

⊂

ℝ

n

that

satisfies

the following:

•

C

is convex;

•

C

is closed;

•

C

is solid, meaning it has nonempty interior;

•

C

is pointed, meaning

x

,

-

x

∈

C

⇒

x

=

0

.

A proper cone

C

induces a

partial ordering

on

ℝ

n

:

a

⪯

b

⇔

b

-

a

∈

C

.

This

ordering

has many nice properties, such as

transitivity

,

reflexivity

, and antisymmetry.

References

1

S. Boyd, L. Vandenberghe,

Convex Optimization

, Cambridge University Press, 2004.

Title

proper cone

Canonical name

ProperCone

Date of creation

2013-03-22 14:37:13

Last modified on

2013-03-22 14:37:13

Owner

dooder0001 (4288)

Last modified by

dooder0001 (4288)

Numerical id

7

Author

dooder0001 (4288)

Entry type

Definition

Classification

msc 52A20

Related topic

Cone3

Related topic

Cone5

Generated on Fri Feb 9 22:15:46 2018 by

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Radon’s lemma

Radon’s lemma

Every set

A

⊂

ℝ

d

of

d

+

2

or more points can be partitioned into two

disjoint sets

A

1

and

A

2

such that the

convex hulls

of

A

1

and

A

2

intersect

.

Proof.

Without loss of generality we assume that the set

A

consists of exactly

d

+

2

points which we number

a

1

,

a

2

,

…

,

a

d

+

2

. Denote by

a

i

,

j

the

j

’th

component

of

i

’th vector, and write the components in a matrix as

M

=

[

a

1

,

1

a

2

,

1

…

a

d

+

2

,

1

a

1

,

2

a

2

,

2

…

a

d

+

2

,

2

⋮

⋮

⋱

⋮

a

1

,

d

a

2

,

d

…

a

d

+

2

,

d

1

1

…

1

]

.

Since

M

has fewer rows than columns, there is a non-zero

column vector

λ

such that

M

⁢

λ

=

0

, which is

equivalent

to the

existence

of a solution to the system

0

=

λ

1

⁢

a

1

+

λ

2

⁢

a

2

+

⋯

+

λ

d

+

2

⁢

a

d

+

2

(1)

0

=

λ

1

+

λ

2

+

⋯

+

λ

d

+

2

Let

A

1

be the set of those

a

i

for which

λ

i

is positive, and

A

2

be the rest. Set

s

to be the sum of positive

λ

i

’s. Then by the system (

1

) above

1

s

⁢

∑

a

i

∈

A

1

λ

i

⁢

a

i

=

1

s

⁢

∑

a

i

∈

A

2

(

-

λ

i

)

⁢

a

i

is a point of

intersection

of convex hulls of

A

1

and

A

2

.
∎

Title

Radon’s lemma

Canonical name

RadonsLemma

Date of creation

2013-03-22 13:56:48

Last modified on

2013-03-22 13:56:48

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Theorem

Classification

msc 52A20

Classification

msc 52C07

Generated on Fri Feb 9 22:15:48 2018 by

LaTeXML
recession cone

recession cone

Let

C

be a

convex set

in

ℝ

n

. If

C

is

bounded

, then for any

x

∈

C

, any ray emanating from

x

will

eventually

“exit”

C

(that is, there is a point

z

on the ray such that

z

∉

C

). If

C

is unbounded, however, then there exists a point

x

∈

C

, and a ray

ρ

emanating from

x

such that

ρ

⊆

C

. A

direction

d

in

C

is a point in

ℝ

n

such that for any

x

∈

C

, the ray

{

x

+

r

⁢

d

∣

r

≥

0

}

is also in

C

(a subset of

C

).

The

recession cone

of

C

is the set of all directions in

C

, and is denoted by denoted by

0

+

⁢

C

. In other words,

0

+

⁢

C

=

{

d

∣

x

+

r

⁢

d

∈

C

,

∀

x

∈

C

,

∀

r

≥

0

}

.

If a convex set

C

is bounded, then the recession cone of

C

is pretty useless; it is

{

0

}

. The converse is not true, as illustrated by the convex set

C

=

{

(

x

,

y

)

∣

0

≤

x

<

1

,

y

≥

1

}

∪

{

(

x

,

y

)

∣

0

≤

x

≤

1

,

0

≤

y

≤

1

}

.

Clearly,

C

is not bounded but

0

+

⁢

C

=

{

0

}

. However, if the additional condition that

C

is closed is imposed, then we recover the converse.

Here are some other examples of recession cones of unbounded convex sets:

•

If

C

=

{

(

x

,

y

)

∣

|

x

|

≤

y

}

, then

0

+

⁢

C

=

C

.

•

If

C

=

{

(

x

,

y

)

∣

|

x

|

<

y

}

, then

0

+

⁢

C

=

C

¯

, the

closure

of

C

.

•

If

C

=

{

(

x

,

y

)

∣

|

x

|

n

≤

y

,

n

>

1

}

, then

0

+

⁢

C

=

{

(

0

,

y

)

∣

y

≥

0

}

.

Remark.

The recession cone of a convex set is convex, and, if the convex set is closed, its recession cone is closed as well.

Title

recession cone

Canonical name

RecessionCone

Date of creation

2013-03-22 16:20:24

Last modified on

2013-03-22 16:20:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 52A20

Classification

msc 52A07

Defines

direction of a convex set

Generated on Fri Feb 9 22:15:58 2018 by

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Banach-Mazur compactum

Banach-Mazur compactum

The

Banach-Mazur metric

is a

distance

on the space of all

http://planetmath.org/node/Isomorphism2

isomorphic

Banach spaces

. If

B

1

,

B

2

are

n

-dimensional Banach
spaces, the distance between them is

d

⁢

(

B

1

,

B

2

)

=

ln

⁢

inf

⁡

{

∥

T

∥

⋅

∥

T

-

1

∥

:

T

∈

G

⁢

L

⁢

(

B

1

,

B

2

)

}

.

Then

d

satisfies the

triangle inequality

, and

d

⁢

(

B

1

,

B

2

)

=

0

if
and only if

B

1

and

B

2

are

isometric

. The space of isometry

http://planetmath.org/node/EquivalenceRelation

classes of

n

-dimensional Banach spaces under this metric is a

compact

metric space, known as a

Banach-Mazur compactum

.

Title

Banach-Mazur compactum

Canonical

name

BanachMazurCompactum

Date of creation

2013-03-22 14:55:24

Last modified on

2013-03-22 14:55:24

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

5

Author

bbukh (348)

Entry type

Definition

Classification

msc 52A21

Classification

msc 46B20

Defines

Banach-Mazur metric

Defines

Banach-Mazur distance

Generated on Fri Feb 9 22:16:00 2018 by

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Wulff theorem

Wulff theorem

Definition 1

(Wulff shape).

Let

ϕ

:

R

n

→

[

0

,

+

∞

)

be a non-negative, convex, coercive, positively

1

-

homogeneous function

. We define the

Wulff shape

relative to

ϕ

as the set

W

ϕ

:=

{

x

∈

ℝ

n

:

⟨

x

,

y

⟩

≤

1

for all

y

such that

ϕ

⁢

(

y

)

≤

1

}

(where

⟨

⋅

,

⋅

⟩

is the

Euclidean

inner product

in

R

n

.)

Theorem 1

(Wulff).

Let

ϕ

:

R

n

→

[

0

,

+

∞

)

be a non-negative, convex, coercive,

1

-homogeneous function. Given a regular open set

D

⊂

R

n

we consider the following anisotropic surface energy:

F

ϕ

⁢

(

D

)

=

∫

∂

⁡

D

ϕ

⁢

(

ν

D

⁢

(

x

)

)

⁢

𝑑

σ

⁢

(

x

)

where

ν

D

⁢

(

x

)

is the outer

unit normal

to

∂

⁡

D

, and

σ

is the

surface area

on

∂

⁡

D

.
Then, given any set

D

with the same volume as

W

ϕ

, i.e.

|

D

|

=

|

W

ϕ

|

, one has

F

ϕ

⁢

(

D

)

≥

F

ϕ

⁢

(

W

ϕ

)

.
Moreover if

|

D

|

=

|

W

ϕ

|

and

F

ϕ

⁢

(

D

)

=

F

ϕ

⁢

(

W

ϕ

)

then

D

is a

translation

of

W

ϕ

i.e. there exists

v

∈

R

n

such that

D

=

v

+

W

ϕ

.

Title

Wulff theorem

Canonical name

WulffTheorem

Date of creation

2013-03-22 15:19:50

Last modified on

2013-03-22 15:19:50

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

8

Author

paolini (1187)

Entry type

Theorem

Classification

msc 52A21

Related topic

FinslerGeometry

Defines

Wulff shape

Generated on Fri Feb 9 22:16:02 2018 by

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star-shaped region

star-shaped region

Definition

A subset

U

of a real (or possibly complex)

vector space

is called

star-shaped

if there is a point

p

∈

U

such that the

line segment

p

⁢

q

¯

is

contained

in

U

for all

q

∈

U

. (Here,

p

⁢

q

¯

=

{

t

⁢

p

+

(

1

-

t

)

⁢

q

|

t

∈

[

0

,

1

]

}

.) We then say that

U

is star-shaped with respect to

p

.

In other , a region

U

is star-shaped if there is a point

p

∈

U

such that

U

can be “collapsed” or “contracted”

p

.

0.0.1

Examples

1.

In

ℝ

n

, any

vector subspace

is star-shaped. Also, the

unit cube

and

unit ball

are star-shaped, but the

unit sphere

is not.

2.

A subset

U

of a vector space is star-shaped with respect to all of its points if and only if

U

is convex.

Title

star-shaped region

Canonical

name

StarshapedRegion

Date of creation

2013-03-22 13:34:13

Last modified on

2013-03-22 13:34:13

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

10

Author

matte (1858)

Entry type

Definition

Classification

msc 52A30

Classification

msc 32F99

Defines

star-shaped

Generated on Fri Feb 9 22:16:04 2018 by

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Helly’s theorem

Helly’s theorem

Suppose

A

1

,

…

,

A

m

⊂

ℝ

d

is a family of convex sets, and every

d

+

1

of them have a non-empty

intersection

. Then

⋂

i

=

1

m

A

i

is non-empty.

Proof.

The proof is by

induction

on

m

. If

m

=

d

+

1

, then the statement is

vacuous

. Suppose the statement is true if

m

is replaced by

m

-

1

. The sets

B

j

=

⋂

i

≠

j

A

i

are non-empty by

inductive hypothesis

. Pick a point

p

j

from each of

B

j

. By Radon’s lemma, there is a

partition

of

p

’s into two sets

P

1

and

P

2

such that

I

=

(

conv

⁡

P

1

)

∩

(

conv

⁡

P

2

)

≠

∅

. For every

A

j

either every point in

P

1

belongs to

A

j

or every point in

P

2

belongs to

A

j

. Hence

I

⊆

A

j

for every

j

.
∎

Title

Helly’s

theorem

Canonical

name

HellysTheorem

Date of creation

2013-03-22 13:57:38

Last modified on

2013-03-22 13:57:38

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Theorem

Classification

msc 52A35

Generated on Fri Feb 9 22:16:06 2018 by

LaTeXML
convex functions lie above their supporting lines

convex functions lie above their supporting lines

Let

f

:

𝐑

→

𝐑

be a convex, twice

differentiable
function

on

[

a

,

b

]

. Then

f

⁢

(

x

)

lies above its supporting lines, i.e. it’s

greater

than any

tangent line

in

[

a

,

b

]

.

Proof.

:

Let

r

⁢

(

x

)

=

f

⁢

(

x

0

)

+

f

′

⁢

(

x

0

)

⁢

(

x

-

x

0

)

be the

tangent

of

f

⁢

(

x

)

in

x

=

x

0

∈

[

a

,

b

]

.

By

Taylor theorem

, with remainder in Lagrange form, one has, for any

x

∈

[

a

,

b

]

:

f

⁢

(

x

)

=

f

⁢

(

x

0

)

+

f

′

⁢

(

x

0

)

⁢

(

x

-

x

0

)

+

1

2

⁢

f

′′

⁢

(

ξ

⁢

(

x

)

)

⁢

(

x

-

x

0

)

2

with

ξ

⁢

(

x

)

∈

[

a

,

b

]

. Then

f

⁢

(

x

)

-

r

⁢

(

x

)

=

1

2

⁢

f

′′

⁢

(

ξ

⁢

(

x

)

)

⁢

(

x

-

x

0

)

2

≥

0

since

f

′′

⁢

(

ξ

⁢

(

x

)

)

≥

0

by
convexity.
∎

Title

convex functions lie above their supporting lines

Canonical name

ConvexFunctionsLieAboveTheirSupportingLines

Date of creation

2013-03-22 16:59:20

Last modified on

2013-03-22 16:59:20

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

5

Author

Andrea Ambrosio (7332)

Entry type

Result

Classification

msc 52A41

Classification

msc 26A51

Classification

msc 26B25

Generated on Fri Feb 9 22:16:11 2018 by

LaTeXML
proof of criterion for convexity

proof of criterion for convexity

Theorem 1

.

Suppose

f

:

(

a

,

b

)

→

R

is

continuous

and that,
for all

x

,

y

∈

(

a

,

b

)

,

f

⁢

(

x

+

y

2

)

≤

f

⁢

(

x

)

+

f

⁢

(

y

)

2

.

Then

f

is convex.

Proof.

We begin by showing that, for any

natural numbers

n

and

m

≤

2

n

,

f

⁢

(

m

⁢

x

+

(

2

n

-

m

)

⁢

y

2

n

)

≤

m

⁢

f

⁢

(

x

)

+

(

2

n

-

m

)

⁢

f

⁢

(

y

)

2

n

by

induction

. When

n

=

1

, there are three possibilities:

m

=

1

,

m

=

0

,
and

m

=

2

. The first possibility is a

hypothesis

of the

theorem

being
proven and the other two possibilities are

trivial

.

Assume that

f

⁢

(

m

′

⁢

x

+

(

2

n

-

m

′

)

⁢

y

2

n

)

≤

m

′

⁢

f

⁢

(

x

)

+

(

2

n

-

m

′

)

⁢

f

⁢

(

y

)

2

n

for some

n

and all

m

′

≤

2

n

. Let

m

be a number less than or equal
to

2

n

+

1

. Then either

m

≤

2

n

or

2

n

+

1

-

m

≤

2

n

. In the
former case we have

f

⁢

(

1

2

⁢

m

⁢

x

+

(

2

n

-

m

)

⁢

y

2

n

+

y

2

)

≤

1

2

⁢

(

f

⁢

(

m

⁢

x

+

(

2

n

-

m

)

⁢

y

2

n

)

+

f

⁢

(

y

)

)

≤

1

2

⁢

m

⁢

f

⁢

(

x

)

+

(

2

n

-

m

)

⁢

f

⁢

(

y

)

2

n

+

f

⁢

(

y

)

2

=

m

⁢

f

⁢

(

x

)

+

(

2

n

+

1

-

m

)

⁢

f

⁢

(

y

)

2

n

+

1

.

In the other case, we can reverse the roles of

x

and

y

.

Now, every

real number

s

has a binary expansion; in other words, there
exists a

sequence

{

m

n

}

of integers such that

lim

n

→

∞

⁡

m

n

/

2

n

=

s

. If

0

≤

s

≤

1

, then we also have

0

≤

m

n

≤

2

n

so, by what we proved above,

f

⁢

(

m

n

⁢

x

+

(

2

n

-

m

n

)

⁢

y

2

n

)

≤

m

n

⁢

f

⁢

(

x

)

+

(

2

n

-

m

n

)

⁢

f

⁢

(

y

)

2

n

.

Since

f

is assumed to be continuous, we may take the limit of both
sides and conclude

f

⁢

(

s

⁢

x

+

(

1

-

s

)

⁢

y

)

≤

s

⁢

f

⁢

(

x

)

+

(

1

-

s

)

⁢

f

⁢

(

y

)

,

which implies that

f

is convex.
∎

Title

proof of criterion for convexity

Canonical

name

ProofOfCriterionForConvexity

Date of creation

2013-03-22 17:00:23

Last modified on

2013-03-22 17:00:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 52A41

Classification

msc 26A51

Classification

msc 26B25

Generated on Fri Feb 9 22:16:14 2018 by

LaTeXML
proof of criterion for convexity II

proof of criterion for convexity II

If

f

was not convex, then there was a point

ξ

∈

(

a

,

b

)

such
that

f

⁢

(

ξ

)

>

h

⁢

(

x

)

=

f

⁢

(

v

)

-

f

⁢

(

u

)

v

-

u

⁢

(

x

-

u

)

+

f

⁢

(

u

)

for some

u

<

v

in

(

a

,

b

)

. Since

f

is

continuous

, there would be a

neighborhood

(

ξ

-

δ

,

ξ

+

δ

)

,

δ

>

0

, of

ξ

such that

f

⁢

(

x

)

>

h

⁢

(

x

)

for all

x

in this neighborhood. (I.e.,

f

⁢

(

x

)

was “above” the

line segment

joining

f

⁢

(

u

)

and

f

⁢

(

v

)

.) Let

s

=

ξ

-

δ

,

t

=

ξ

+

δ

.

Using the two points

A

=

(

s

,

f

⁢

(

s

)

)

,

B

=

(

t

,

f

⁢

(

t

)

)

, we construct
another line segment

A

⁢

B

¯

whose equation is given by

g

⁢

(

x

)

=

f

⁢

(

s

)

-

f

⁢

(

t

)

2

⁢

δ

⁢

(

x

-

s

)

+

f

⁢

(

s

)

; we have

f

⁢

(

x

)

>

g

⁢

(

x

)

for

x

∈

(

s

,

t

)

. In particular,

f

⁢

(

ξ

)

=

f

⁢

(

s

+

t

2

)

>

g

⁢

(

ξ

)

=

f

⁢

(

s

)

+

f

⁢

(

t

)

2

.

(1)

(One easily verifies

g

⁢

(

ξ

)

=

(

f

⁢

(

s

)

+

f

⁢

(

t

)

)

/

2

.) This contradicts

hypothesis

.

Note that we have tacitly used the fact that

h

⁢

(

x

)

=

λ

⁢

f

⁢

(

v

)

+

(

1

-

λ

)

⁢

f

⁢

(

u

)

for some

λ

and

g

⁢

(

x

)

=

λ

⁢

f

⁢

(

s

)

+

(

1

-

λ

)

⁢

f

⁢

(

t

)

for some

λ

.

Title

proof of criterion for convexity II

Canonical name

ProofOfCriterionForConvexityII

Date of creation

2013-03-22 18:25:25

Last modified on

2013-03-22 18:25:25

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

5

Author

yesitis (13730)

Entry type

Proof

Classification

msc 52A41

Classification

msc 26A51

Classification

msc 26B25

Generated on Fri Feb 9 22:16:17 2018 by

LaTeXML
convex set

convex set

Let

S

a subset of

ℝ

n

. We say that

S

is

convex

when, for any pair of points

A

,

B

in

S

, the segment

A

⁢

B

¯

lies entirely inside

S

.

The former statement is

equivalent

to saying that for any pair of vectors

u

,

v

in

S

, the vector

(

1

-

t

)

⁢

u

+

t

⁢

v

is in

S

for all

t

∈

[

0

,

1

]

.

If

S

is a

convex set

, for any

u

1

,

u

2

,

…

,

u

r

in

S

, and any positive numbers

λ

1

,

λ

2

,

…

,

λ

r

such that

λ

1

+

λ

2

+

⋯

+

λ

r

=

1

the vector

∑

k

=

1

r

λ

k

⁢

u

k

is in

S

.

Examples of convex sets in the plane are circles,

triangles

, and ellipses.
The definition given above can be generalized to any

real vector space

:

Let

V

be a

vector space

(over

ℝ

or

ℂ

). A subset

S

of

V

is

convex

if for all points

x

,

y

in

S

, the

line segment

{

α

⁢

x

+

(

1

-

α

)

⁢

y

∣

α

∈

(

0

,

1

)

}

is also in

S

.

More generally, the same definition works for any vector space over an

ordered field

.

A

polyconvex set

is a finite union of compact, convex sets.

Remark

. The notion of convexity can be generalized to an arbitrary

partially ordered set

: given a poset

P

(with

partial ordering

≤

), a subset

C

of

P

is said to be

convex

if for any

a

,

b

∈

C

, if

c

∈

P

is between

a

and

b

, that is,

a

≤

c

≤

b

, then

c

∈

C

.

Title

convex set

Canonical name

ConvexSet

Date of creation

2013-03-22 11:46:35

Last modified on

2013-03-22 11:46:35

Owner

drini (3)

Last modified by

drini (3)

Numerical id

20

Author

drini (3)

Entry type

Definition

Classification

msc 52A99

Classification

msc 16G10

Classification

msc 11F80

Classification

msc 22E55

Classification

msc 11A67

Classification

msc 11F70

Classification

msc 06A06

Synonym

convex

Related topic

ConvexCombination

Related topic

CaratheodorysTheorem2

Related topic

ExtremeSubsetOfConvexSet

Related topic

PropertiesOfExtemeSubsetsOfAClosedConvexSet

Defines

polyconvex set

Defines

polyconvex

Generated on Fri Feb 9 22:16:19 2018 by

LaTeXML