Approximating Convex Bodies by Cephoids

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Oktober 2020 640

Approximating Convex Bodies by Cephoids

Joachim Rosenm¨ uller

Center for Mathematical Economics (IMW) Bielefeld University

Universit¨ atsstraße 25 D-33615 Bielefeld · Germany e-mail: imw@uni-bielefeld.de http://www.imw.uni-bielefeld.de/wp/

ISSN: 0931-6558

This work is licensed un- der a Creative Commons

“Attribution 4.0 Interna-

tional” license.

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(0, x e ₂ , x e ₃ ) + Π e ⁽¹⁾ = (0, e a ⁽²⁾ ₂ , e a ⁽³⁾ ₃ ) + Π e ⁽¹⁾ = Π e ⁽¹⁾ + e a ⁽²⁾² + a e ⁽³⁾³ .

2

Π e ⁽¹⁾

⁽

Π e ⁽²⁾

Π ⁽³⁾

^#

$ 5) (

6) /

, A

#

>1

e

x ⁽⁻²⁾ + Π e ⁽²⁾ = a e ⁽¹⁾¹ + Π e ⁽²⁾ + a e ⁽³⁾³ .

, A

#

>G1

e

x ⁽⁻³⁾ + Π e ⁽³⁾ = a e ⁽¹⁾¹ + a e ⁽²⁾² + Π e ⁽³⁾ .

$

, A

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>1

4 , A

#

>1

4 , A

#

>G1

) )

Π e

^# ^;6)

(

, A

#

>D1

e

x = ( x e 1 , x e 2 , x e 3 ) = a e ⁽¹⁾¹ + e a ⁽²⁾² + a e ⁽³⁾³ .

(15)

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) # 2)4

e

x

⁾ ^.) ^/

)5)

Π e

^# ^{& 4}⁵⁽⁽ ^,^A^#^>A1⁴⁽⁶

n ^x ^e

x e ⁽⁻ⁱ⁾ + Π e ⁽ⁱ⁾ (i = 1, 2, 3)

x e

^# ² ⁾

(a) x e

Π, e (b) n ^x ^e

Π e

x e .

, A

#

>E1

◦ eeeeee ^◦

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5)

6

) 5

2

x e

) (

Π ⁽ ^x) ^e

n ^x ^e

%

Γ

x e

^# ² ⁾

∂ Π e

^,⁽

1

#

,&$ $" +.,( $. +$% &$!.$

01

Π = Π e ^(e ^x) = P

i∈I Π e ⁽ⁱ⁾

^"^! ^! ^!

x e

e

x

^"

Π ⁽ ^x) ^e

^!

n ^x ^e

"

e

x

^"

Π ⁽ x) e

^" ^!^"

• eee ^◦ eee ^•

2

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F

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:

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#

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i ∈ I

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#

>H1

e

K ⁱ = K ^e ^a ⁽ⁱ⁾ⁱ

Π e ⁽ⁱ⁾

a e ⁽ⁱ⁾ⁱ

^# ²

Π e ⁽ⁱ⁾

^# ²

K e ⁱ

⁽

, A

#

)?1

X

j∈i∈I\{i}

e

a ^(j)j + Π e ⁽ⁱ⁾

X

j∈I

e

a ^(j)j = x e ;

7)

(5

< ' A

#

4 C

) , A

# E1

#

2

P

j∈i∈I\{i} a e ^(j)j + Π e ⁽ⁱ⁾

⁶

Γ

n

(

∂Γ

X

j∈i∈I\{i}

e

a ^(j)j + a e ^(i)l (l ∈ i ∈ I \ {i})

X

j∈I

e

a ^(j)j = x e

, A

# )>1

% 7)

4

K e ⁱ

n ^x ^e

∂Γ

x e

i ∈ I

^# ^$⁾

, #5#42

>

#

4

=

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4

, A

#

))1

x e = X

j∈I

e

a ^(j)j

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?

Π e

^# ⁴ ⁾

n

ⁿ ₊

(

4

x e

^#

2

e n = n ^(e ^x)

e a ⁽ⁱ⁾ⁱ

Π e ⁽ⁱ⁾ (i ∈ I)

^#

.) ,

)( 1

(#

x e ∈ ∂Γ

Π ⁽⁰⁾ = Π ^a ⁽⁰⁾

⁶ ^.)^{/ #}

e n = n ^(e ^x)

x e

Π e

K ^(0)j

⁶

∆ ⁽⁰⁾

a ^(0)j

^# ^*

, A

# )A1

I ⁽⁰⁾ _{ _x} _e :=

j ∈ I e n ∈ K ^(0)j .

2

4(

a ^(0)j

⁽ ⁽

e n

x e

∆ e

^#

2 (

x e ∈ ∂Γ

^#

Π = Π e ^(e ^x)

⁶ ⁾⁽⁵

6

x e

Π ⁽⁰⁾

⁶ ^.) ^{/ #} ²

, A

# )1

I ⁽⁰⁾ _{e _x} 6= ∅

) #

• eee ^◦ eee ^•

2

*

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5)

4

# 2

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=

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# 2

K ^(0)j (j ∈ I )

⁾

ⁿ ₊

^#

( 6

(

)

) (

)

∂Γ

^#

x e ∈ ∂ Γ

Π = Π e ^(e ^x) = P

i∈I Π e ⁽ⁱ⁾

^"^!

!

x e

i ∈ I

^"^"

l ∈ I \ {i}

^"

x e ^(−i+l)

^"

Π e

x e ^(−i+l) ∈ ∂Γ ⁽⁻ⁱ⁾ = ∂Γ ∩ {x x i = 0}

e n

| ^I\{i}

"

Π e x e ^(−i+l)

2

4

∂ Γ

Π e

6)

∂ Γ ⁽⁻ⁱ⁾

^# ²⁵ ⁽

x e

^{( 4}⁴^*

n + 1

(17)

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∂Γ

Π e

⁶ ⁶ ^# ⁸

) 4

Π e

⁽⁴ ^.)

Π e ⁽ⁱ⁾ (i ∈ I )

∂Γ

^#

C

i ∈ I

⁶

x e ⁽⁻ⁱ⁾ := x e

| ^I−{i}

# 2

, A

#

)1

x e ⁽⁻ⁱ⁾ = X

l∈I\{i}

a ^(l)l .

2

e n ⁽⁻ⁱ⁾ := n ^x ^e

| ^I ^\{i}

a ^(l)l

∆ ^a ^(l)

l ∈ I \ {i}

⁴

e

x ⁽⁻ⁱ⁾

^%

, A

#

)G1

X

l∈I\{i}

∆ ^a ^(l) | ^I−{i}

2

K ^(i)l (l ∈ I \ {i})

a ^(i)l

∆ ^a ⁽ⁱ⁾

⁾

ⁿ _I\{i}

⁴^{# #4}

, A

# )D1

I ⁽⁻ⁱ⁾ _x _e := {l l ∈ I \ {i}, e n ⁽⁻ⁱ⁾ ∈ K ^(i)l } 6= ∅ .

2

4

c

⁾

, A

# )E1

C ⁽⁻ⁱ⁾ _x _e := CovH {a ^(i)l (l ∈ I ⁽⁻ⁱ⁾ _x _e )}

6

(

a ^(l)l (l ∈ I \ {i})

^# ^{9 4}⁽

, A

# )1

(

*

, A

#

)H1

x e ⁽⁻ⁱ⁾ + c = X

l∈I\{i}

a ^(l)l + c

3

, A

#

A?1

X

l∈I\{i}

∆ â ^(l) | Î−{i} + ∆ â ⁽ⁱ⁾ | ^{I ^−{i}} = Π e | ^{I−{i}} = Π e ⁽⁻ⁱ⁾ .

) 4

C ⁽⁻ⁱ⁾ _x _e

, A

# A>1

e

x ^(−i+l) = x e ⁽⁻ⁱ⁾ + a ^(i)l (l ∈ I ⁽⁻ⁱ⁾ _x _e ) .

$

∂ Γ ⁽⁻ⁱ⁾

⁶ ⁾^# ^%

e n ⁽⁻ⁱ⁾

Π e ⁽⁻ⁱ⁾

Π e

^#

6

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6

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3

^# ^{9 4}

Π e

⁶

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0

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3 +

#

)

B )

# )

4

-

)

2 0

, 6 =

>?@1

#

2

%

Π e

⁴

AD

(18)

?

e

x 1 x e 2

e x 3

h ² h ¹

h ³

e x ⁽⁻²⁺³⁾

e x ⁽⁻²⁺¹⁾

e x ⁽⁻¹⁺²⁾ e

x ⁽⁻¹⁺³⁾ e x ⁽⁻¹⁾

f ¹

f ² f ³

e

x x e ^(−1−3+2)

Π ⁽³⁾

e x ⁽⁻³⁺¹⁾

e x ⁽⁻³⁺²⁾

C

5) A

# 2

2 3 )

% C

5) A

#

(

C

5) )

# A

# 2

x e

6)

x e ^(−i+l)

Π e

^# ^<

e

x = ∆ ê â _{2} ⁽¹⁾ + ∆ ê â _{3} ⁽²⁾ + ∆ ê â _{1} ⁽³⁾

(

3

)

4

( 6

Λ â _{12}{13} ê ⁽¹⁾ â ê ⁽²⁾

Λ ê â _{13}{12} ⁽²⁾ ê â ⁽³⁾

( .) /

∆ ^e ^a _{123} ⁽¹⁾

∆ ^a _{123} ^e ⁽³⁾ .

,

.) /

1

∆ ^e ^a ⁽²⁾

^#

('

2 A

# H

I ⁽⁻³⁾ _x _e = {1}

⁾

e

x ⁽⁻³⁺¹⁾

∂Γ

Π e

∂Γ ⁽⁻³⁾

^#

◦ eeeeee ^◦

(

( (

)

) (

-

( 0

4

4 ) (

*

(##

#

Γ x _b

b x ∈ ∂Γ

^#

x b ∈ Γ, x e ∈ ∂Γ

⁶ ⁾

x b < x e

^# ² ^{(! ,(}

/

"- 0/

((

Π ⁽ _x _b ^x) ^e

⁾⁽⁵ ⁶

x e − x b

^*^A^#

(

Γ x _b

#

• eeeeee ^•

C

5) A

#

5

(5 ) # C

(

Γ x _b = Γ ∩ {x ≥ x} b x b

^5# ² ⁽ ⁾ ⁾

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?

e x b

x

∧

e x ⁽⁻ⁱ⁺¹⁾

∧

e x ⁽⁻ⁱ⁺³⁾ b

x + ( x e − x) b ⁽⁻ⁱ⁾

∂Γ

∧

Π e ⁽ⁱ⁾

C

5) A

# 2

2

Π ^(e _x _b ^x)

((##

#

x e − x b ∈ ∂Γ x _b

# C

4

)( 6 '

5

∂Γ

^# ² ⁽

, A

#

A )1

x b + Π ^(e _x _b ^x)

Γ

^#

2 ( )

n

^.)^/

, A

#

AA1

Π ⁽ _x _b ^x) ^e =: X

i∈I

Π

∧

e a ⁽ⁱ⁾ ;

( (

, A

#

A1

Π

∧

e a ⁽ⁱ⁾

=:

∧

Π e ⁽ⁱ⁾

2

∧

e

a ⁽ⁱ⁾

⁶ ^* ^A^# ^, ⁾ ^,^A^#¹¹ ^"

!

"# 2 5 6

6

Γ x _b

5)

< '

A

#

)

( , A

# E1

, A

# A 1

∧

e

a ⁽ⁱ⁾ _i = x e _i − x b _i (i ∈ I )

4 5)

, A

# H1

, A

#

AG1

x e = x b + ( ^∧ e a ⁽¹⁾ ₁ , . . . , e ^∧ a ⁽ⁿ⁾ _n ) = x b + X

i∈I

∧

e a ⁽ⁱ⁾ⁱ .

) # 2

4 )

5 (

x b

⁴

x e

⁾

.) /

Π

∧

e

a ⁽ⁱ⁾ (i ∈ I )

^#

2

.) /

, A

#

AD1

x b + ( x e − x) b ⁽⁻ⁱ⁾ + e ∧

Π ⁽ⁱ⁾ = x b + X

j∈I\−{i}

∧

e a ^(j)j +

e ∧

Π ⁽ⁱ⁾ (i ∈ I )

(20)

?

(

, A

#

AE1

x e

⁽

x b + ( x e − x) b ⁽⁻ⁱ⁾ + e a ^∧ j e ^j := x ^∧ e ^(−i+j) (j ∈ I)

5

, A

#

>>1

# C

5) A

#

(

)

.) /

e ∧

Π ⁽ⁱ⁾

⁾ ^{6) #} ² ^*⁵⁾ ⁾

(

) ( (

#

$5 ( +

n ^x ^e

∂Γ x _b

x e

⁽

n ^x ^e

∂Γ

x e

^# ^$ ^.)^/

∧

Π e ⁽ⁱ⁾

⁶

Γ x _b

4( )

>

# C

i ∈ I

n ^x ^e

e ∧

Π ⁽ⁱ⁾

a ^∧ e ⁽ⁱ⁾ⁱ

^# ² ⁾ ²

n ^x ^e

, A

#

AH1

x b + X

j∈I\−{i}

∧

e a ^(j)j +

∧

Π e ⁽ⁱ⁾

x b + X

j∈I

∧

e

a ^(j)j = x e .

)

# % 7)

4

, A

#

?1

x b + X

j∈I

∧

e

a ^(j)j = x e ;

∧

Π e ⁽ⁱ⁾ (i ∈ I )

^#

A

# 2)4

e

x

∂Γ

⁽

x b + Π ^(e _x _b ^x)

^#

$4

n ^x ^e

∂Γ

Π ⁽ _x _b ^x) ^e + x b

x e

^#

• eee ^◦ eee ^•

C

(

2 A

# D

, A

#

>E1

#

Approximating Convex Bodies by Cephoids

Oktober 2020 640