Extremal Problems Related to Dual Gauss John Position

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ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2018.612216 Dec. 26, 2018 2589 Journal of Applied Mathematics and Physics

Extremal Problems Related to Dual Gauss-John

Position

Tongyi Ma

College of Mathematics and Statistics, Hexi University, Zhangye, China

Abstract

In this paper, the extremal problem, min{ (lp

φ

K o) : ∈

φ

K L⊆ ,

φ

∈GL( )}n , of

two convex bodies K and L in _n_{is considered. For K to be in extremal}

po-sition in terms of a decompopo-sition of the identity, give necessary conditions together with the optimization theorem of John. Besides, we also consider the weaker optimization problem:

1 2

min{( ( )) :p n, n , GL( )}

p

l

φ

K

φ

K B_⊆

φ

K S_∩ − _{≠ ∅ ∈}

φ

n

. As an application, we give the geometric distance between the unit ball B2n and a centrally

sym-metric convex body K.

Keywords

Dual Gauss-John Position, Optimization Theorem of John, Dual lp-Norm,

Contact Pair

1. Introduction

Let γn be the classical Gaussian probability measure with density

2 | |

2 1 ( 2 )

x

ne

π −

,

and || ||⋅ K is the Minkowski functional of a convex body K⊂n. An

impor-tant quantity on local theory of Banach space is the associated l-norm:

( ) n|| || d ( ).K n

l K =

_∫

x γ x



The minimum of the functional

d ( ) n xφK γn x

∫



under the constraint φK⊆B2n is attained for φ =In, then a convex body K is

in the Gauss-John position, where φ ∈GL( )n _, _B₂n_{is the Euclidean unit ball}

How to cite this paper: Ma, T.Y. (2018) Ex-tremal Problems Related to Dual Gauss-John Position. Journal of Applied Mathematics and Physics, 6, 2589-2599.

https://doi.org/10.4236/jamp.2018.612216

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DOI: 10.4236/jamp.2018.612216 2590 Journal of Applied Mathematics and Physics

and In is the identity mapping from n to n.

For _x_∈_n\{ }_o _{, the map}_{x x}_⊗ _:_n _→_n_{is the rank 1 linear operator}

,

y_〈x y x〉 _.

Giannopoulos et al. in [1] showed that if K is in the Gauss-John position, then

there exist m n n≤ ( +1) / 2_{contact points} 1

1, , ,2 m n

x x x _{∈ ∂ ∩}K S −

 , and

con-stants c c1, , ,2  cm >0 such that

1 1

m i i

c

= =

∑

and

1

( ) || || d ( ) || || d ( ) .

n n

m

n K n K n i i i

i

x x I x γ x x γ x c x x

=

 

⊗ − = _ ⊗ _



∑



∫



∫



Note that the Gauss-John position is not equivalent to the classical John tion. Giannopoulos et al. [1] pointed out that, when K is in the Gauss-John posi-tion, the distance between the unit ball B2n and the John ellipsoid is of order

/ log n n.

Notice that the study of the classical John theorem went back to John [2]. It states that each convex body K contains a unique ellipsoid of maximal volume, and when B2n is the maximal ellipsoid in K, it can be characterized by points of

contact between the boundary of K and that of B2n. John’s theorem also holds

for arbitrary centrally symmetric convex bodies, which was proved by Lewis [3] and Milman [4]. It was provided in [5] that a generalization of John’s theorem for the maximal volume position of two arbitrary smooth convex bodies. Bastero and Romance [6] proved another version of John’s representation removing smoothness condition but with assumptions of connectedness. For more infor-mation about the study of its extensions and applications, please see [7]-[13].

Recall that a convex body K is a position of K if K =φK a+ _{, for some} non-degenerate linear mapping φ ∈GL( )n and some _a_∈_n_{. We say that K is}

in a position of maximal volume in L if K L⊆ _{and for any position} K of K such that K⊆L_{we have} vol ( ) vol ( )n K ≤ n K , where vol ( )n ⋅ denotes the

vo-lume of appropriate dimension.

Recently, Li and Leng in [14] generalized the Gauss-John position to a general situation. For p≥1_{, denote} l_p-norm by

(

)

1

||

( ) n ||p d ( ) .p

p x K n

l K =

_∫

γ x

 (1.1)

They consider the following extremal problem:

{

}

min lp(φK o) : ∈φK L⊆ ,φ∈GL( ) ,n (1.2)

where L is a given convex body in _n_and_{K is a convex body containing the}

origin o such that o K L∈ ⊆ .

Li and Leng [14] showed that let L be a given convex body in _n_{and K be a}

convex body such that o K L∈ ⊆ . If K is in extremal position of (1.2), then there exist _{m n}_≤ 2 _{contact pairs}

1

( , )x yi i ≤ ≤i m of ( , )K L , and constants

1, , , ,2 m 0

c c _{ }c > _{such that}

1 1

( )d ( ) , 1,

n

m m

n i i i i

i i

I x x µ x p c x y c

= =

=

_∫

⊗ −

∑

⊗

∑

=

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where d ( )µ x is the probability measure on _n_{with normalized density}

d ( ) || || d ( ) / ( ( )) .p p

K n p

x x x l K

µ

=

γ

In this paper, we first present a dual concept of lp-norm l Kp( ). The

genera-lizations of John’s theorem and Li and Leng [14] play a critical role. It would be impossible to overstate our reliance on their work.

For p≥1, we define the dual lp-norm of convex body K by

(

)

1

( ) n ( ) d ( ) ,p p

p K n

l K =

_∫

ρ x γ x



 _(1.3)

where ρK is the radial function of the star body K about the origin.

Now, we consider the extremal problem:

{

}

min lp(φK o) : ∈φK⊆L,φ∈GL( ) ,n (1.4)

where L is a given convex body in _n_and_{K is a convex body containing the}

origin o such that o K L∈ ⊆ _.

Then we prove that the necessary conditions for K to be in extremal position in terms of a decomposition of the identity.

Theorem 1.1. Let L be a given convex body in _n_and_{K be a convex body}

such that o K L∈ ⊆ _{. If K is in extremal position of (1.4), then there exist} 2

m n≤ _{contact pairs} ( , )x yi i ₁≤ ≤i m of ( , )K L , and c c1, , ,2  cm >0 such that

1 1

( )d ( ) , 1,

n

m m

n i i i i

i i

I x x µ x p c x y c

= =

=

_∫

⊗ −

∑

⊗

∑

=

 

where d ( )µ x is the probability measure on _n_{with normalized density}

d ( ) || || d ( ) / ( ( )) .p p

K n p

x x x l K

µ

_ ₌ −

γ



Next the following result is obtained, which is an restriction that is weaker than the extremal problem (1.4):

(

)

{

1

}

2

min ( ) :p n, n , GL( ) .

p

l φK φK_⊆B φK S_ − _{≠ ∅ ∈}φ n _(1.5)

Theoren 1.2. Let K be a given convex body in _n_{. If} n

I is the solution of the extremal problem (1.5), then there exist contact points u u, ′_of_{K and} _B₂n such that

(

)

2 1 2

, ( ) n|| || o( ), , ( ) , ,

p _p

p K _K n

u_′θ _≤ l K

_∫

x − − _∇h x θ xθ γd x _≤ uθ



 _(1.6)

for every _θ_∈_Sn−1_.

The rest of this paper is organized as follows: In Section 2, some basic nota-tion and preliminaries are provided. We prove Theorem 1.1 and Theorem 1.2 in Section 3. In particular, as an application of the extremal problem of

(

)

{

2

}

min ( ) :p n, GL( ) ,

p

l φK o∈φK⊆B φ∈ n _(1.7)

Section 3 shows the geometric distance between the unit ball B2n and a

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2. Notation and Preliminaries

In this section, we present some basic concepts and various facts that are needed in our investigations. We shall work in _n_{equipped with the canonical}

Eucli-dean scalar product 〈⋅ ⋅〉, _{and write}| |⋅ _{for the corresponding Euclidean}

norm. We denote the unit sphere by _Sn−1_.

Let K be a convex body (compact, convex sets with non-empty interiors) in

n

 . The support function of K is defined by

( ) max{ , : }, n.

K

h x = 〈x y y K〉 ∈ x∈

Obviously, ( ) ( t )

K K

h xφ =h

φ

x for φ ∈GL( )n , where φt denotes the trans-pose of

φ

.

A set _K_⊂_n_{is said to be a star body about the origin, if the line segment}

from the origin to any point x K∈ _{is contained in K and K has continuous and} positive radial function ρ ⋅K( ). Here, the radial function of K,ρK:Sn−1→[0, )∞ , is defined by

( ) max{ : }.

K u u K

ρ = λ λ ∈

Note that if K be a star body (about the origin) in _n_{, then K can be uniquely}

determined by its radial function ρ ⋅K( ) and vice verse. If α >0, we have

1

( ) ( )

K x K x

ρ α ₌α ρ− _and _{( )} _{( ).}

K x K x

α

ρ =αρ

More generally, from the definition of the radial function it follows imme-diately that for φ ∈GL( )n the radial function of the image φK={ :φy y K∈ }

of star body K is given by _{( )} ₍ 1 ₎

K x K x

φ

ρ

₌

ρ φ

− _{, for all} _x_∈_n_.

If , n

o

K L∈ and

λ µ

, ≥0 (not both zero), then for p>0, the Lp-radial

combination, n

p o

K L

λ

+

µ

∈ , is defined by (see [15])

( , )p ( , )p ( , ) .p

p

K L K L

ρ λ

+

µ

⋅ =

λρ

⋅ +

µρ

⋅ (2.1)

If a star body K contains the origin o as its interior point, then the Minkowski functional || ||⋅ K of K is defined by

|| ||x K=min{λ>0 :x∈λK}.

In this case,

1

|| ||x K=ρK− ( )x =h x_K°( ),

where K°

denotes the polar set of K, which is defined by

{ n: , 1 for all }. K°₌ x_∈ _〈x y_{〉 ≤} y K_∈



It is easy to verify that for φ ∈GL( )n ,

(_φ_K)°₌_φ−t_K°,

where _φ−t_{denotes the reverse of the transpose of}

_φ

_{. Obviously,}_{( )}_K° ° ₌_K

(see [13] for details).

Let K and L be two convex bodies in _n _{. According to [4], if}

n

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1) x K∈ ∩ ∂L, 2) y L_{∈ ∩ ∂}° K°_,

3) 〈x y, 〉 =1.

If _{x y}, _∈_n_{, we denote by}_{x y}_⊗ _{the rank one projection defined by}

( ) ,

x y u⊗ = 〈x u y〉 _{for all} _u_∈_n_.

The geometric distance δG( , )K L of the convex bodies K and L is defined by

( , ) inf{ : 0, 0,(1/ ) }.

G K L L K L

δ = αβ α > β > β ⊂ ⊂α

3. Proof of Main Results

First, we prove that lp( )⋅ is a norm with respect to Lp-radial combination in n

o

 . Apparently, l Kp( ) 0≥ and l Kp( ) 0= if and only if K =

{ }

o . At the

same time, l cKp( )=cl Kp( ) if real constant c>0. In addition, it is follows

that

( ) ( ) ( ).

p p p p

l K + L ≤l K l L +

Indeed, we have

(

)

(

)

(

) (

)

1

1 1

( ) ( )d ( )

( )d ( ) ( )d ( )

( ) ( ).

n p

n n

p p

p p K L n

p p p

K n L n

p p p p

K n L n

p p

l K L x x

x x x x

l K l L

ρ γ

ρ γ ρ γ

+

+ =

= +

≤ +

= +

∫

 

 

 _

 

Therefore, lp( )⋅ is a norm with respect to Lp-radial combination and on is normed space for lp( )⋅ .

Now, we prove the optimization theorem of John [2] (see [10] also).

Lemma 3.1. Let _:_N _→__{be a}_C1_{-function. Let S be a compact metric}

space and _:_N_{× →}_S __{be continuous. Suppose that for every}_{s S}_∈ _,

( , )

z z s

∇ exists and is continuous on _N_×_S_.

Let _₌{_z_∈_N: ( , ) 0, for all_ _{z s} _≥ _{s S}_∈ }_and 0

z ∈ satisfy

0

( ) minz = z∈ ( ).z

 

Then, either ∇z( ) 0z0 = , or, for some 1≤m N≤ , there exist 1, , ,2 m

s s _ s ∈S _and λ λ₁, , ,₂ λ ∈m  such that ( , ) 0,z s0 i = λi ≥0 for

1≤ ≤i m_{, and}

0 0

1

( ) m ( , ).

z i z i

i

z λ z s

= ∇  =

∑

∇

Using a similar argument as that in [1], we give the proof of Theorem 1.1. Proof of Theorem 1.1. For _{N n}₌ 2_{, we define} __:_N _→__by

(

)

1

( )φ =lp(φK)=

∫

_n||φ−x|| d ( ) ,−Kp γn x p

 (3.1)

where _φ_∈_N_{is the linear mapping from} _n_to _n_{. Clearly} _ _is _C1_{. For}

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DOI: 10.4236/jamp.2018.612216 2594 Journal of Applied Mathematics and Physics ( ,( , )) 1φ x y = − 〈φx y, .〉



The set

{_z N: ( , ) 0,_{z s} _{s S}}

= ∈ ≥ ∈

 

is just the set of elements _φ_∈_N_{such that}

_φ

_{K L}_⊆ _{. If K is in extremal} posi-tion of min{ (lp

φ

K o) : ∈

φ

K L⊆ ,

φ

∈GL( )}n , then  attains its minimum on

 at In, namely,

( )In =l Kp( ) min{ (= lp

φ

K o) : ∈

φ

K L⊆ ,

φ

∈GL( )}.n



Now we prove ∇φ( )In . It follows from (3.1) that

(

)

2 2 1 1 1 | | 1 2 2 1 | | 2 2

( ) || || d ( )

(2 ) || || d

(2 ) (det ) || || d .

n n n p p K n p n x p K p n x p K x x

x e x

x e φ x

φ φ γ

π φ π φ − − − ₋ ₋ − − ₋ − =     =         =    

∫

   

It is easy to obtain that for non-degenerate

φ

, we have

( ,( , ))x y x y, x y, x y

φ φ φ φ φ φ

∇ = −∇ 〈 〉 = ∇ 〈 ⊗ 〉 = − ⊗ and 2 2 2 1 | | 2 | | 1 2 | | 2 1

( ) (2 ) (det ) || || d

(2 ) (det )( ) || || d

(2 ) (det ) || || d ,

n n n q x n p x K x n p _x K x n p x K

x e x

p

x e x

x e x x x

φ

φ π φ

π φ φ

π φ − − − ₋ − − _{− ∗} ₋ − − ₋     ∇ =       ×    − ⊗ 

∫

   

where 1 1 1_{p q}+ = _, _{( )}_φ−1 *_{denotes conjugate of transposed transformation of} 1

φ− _{, and} _φ−1_{is inverse transform of}_{φ ∈}_{GL( )}_n _.

Since  attains its minimum on  at z0=In, combining with Lemma

3.1, it follows that for some m N≤ _{, there exist}λ ≥i 0, s Si∈ , si =( , )x yi i ,

1≤ ≤i m, such that

, 1 ( ,( , )) 1, 1 ,

i i n i i

x y I x y i m

〈 〉 = −_ = ≤ ≤ and

(

)

1 1 1

( ) ( ) ( ) || || d ( )

( ,( , )) . n p p q

n p n K n

m

i n i i

i m

i i i

i

I l K I x x x x

p

I x y

x y φ φ γ λ λ − ₋ = = ∇ = − ⊗ = ∇ = − ⊗

∫

∑

  

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From x yi, i =1,x K L y Li∈ ⊆ , i∈ ⊆ K, we yield xi∈ ∂L and yi∈ ∂K. Taking the trace in (3.2), we have

(

)

(

)

(

)

(

)

(

)

2 2 1 2

1 2 1 2

0 0

Tr ( )

1

Tr ( ) ( ) || || d ( )

1 _{( )} _{|| || d ( )} _{| | || || d ( )}

1 _{( )} _d _d _{|| || d ( )}

1 _{( )} _||

n n n n n p p q

p n K n

p

p p

q

p K n K n

r r

p

n p n p p

q

p _S K

p q p

I

l K I x x x x

p

l K n x x x x x

p

l K n r e r r e r S

p

l K p x

p φ γ γ γ θ θ − − ₋ − ₋ ₋ − − − ∞ _{− −} ∞ _{− +} ₋ − ∇   = _ − ⊗ _     = _ − _     = −     =

∫

       

(

∫

_n || d ( )K−p γn x

)

=l Kp( ).

Suppose

λ

i =c l Ki p ( ). Together with (3.2), we obtain

1

( ) || || d ( ) ( ( )) ( ),

n

m

p p

n K n p i i i

i

x x I x − γ x p l K c x y

= ⊗ − =

∑

⊗

∫

  where 1 1 m i i= c

=

∑

. This completes the proof.  If L B= 2n and ( , ) 1 |φ x = − φx|2, then using the same method in the proof

of Theorem 1.1, we obtain

Corollary 3.2. Let K be a convex body such that o K∈ ⊆B2n. If K is in

ex-tremal position of (1.7), then there exist contact points 1

1, , ,2 m n

u u u _{∈ ∂ ∩}K S −



with _{m n}_≤ 2_and

1, , ,2 m 0

c c _ c > _{, such that,}

1 1

( )d ( ) , 1,

n

m m

n i i i i

i i

I x x µ x p c u u c

= =

=

_∫

⊗ −

∑

⊗

∑

=

 

d ( ) || || d ( ) / ( ( )) .p p

K n p

x x x l K

µ

_ ₌ −

γ



Proof of Theorem 1.2. Suppose that _φ_∈_L( ,_{ }n n)_and _ε_>₀_{is small}

enough. Then

1 1

1: (min_{u S}n ||u u|| )(K In )

φ = _∈ − −εφ −εφ −

satisfies 1

1K B2n, 1K Sn

φ _⊆ φ _∩ − _{≠ ∅}_{. Therefore}

1

|| || d ( ) ( ( )) (min n || || ) .

n x−εφx K−p γn x ≤ l Kp p _{u S}_∈ − u−εφu K −p

∫

 

Let uε be a point on Sn−1 at which the minimum is attained. Observe that

1 2

|| || p || || p || || p ( ), ( )

K K K _K

x−εφx − = x − +εp x − − 〈∇h x x O° φ 〉 + ε

and

2

|_u _u | p 1 _{p u} , _u _O( ). ε −εφ ε − = +ε 〈 ε φ ε〉 + ε

Since _u _Sn1

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(

)

(

)

(

)

(

)

(

)

1 1

|| || ( ), d ( ) ( )

min || || 1

( )

| | 1

( )

( ) , ( ) .

n

p

K _K n

p

p _{u S} K

p

p p

p

p p

p x h x x x O

u u

l K

u u

l K

l K p u u O

ε ε

φ γ ε

εφ ε εφ

ε

φ ε

− − −

−

∈

−

∇ +

− −

≤

− −

≤

= +

∫

_ 



(3.3)

If u is a contact point of K and B2n, then

1+ε φ|| || ||≥ u−εφu|| ||K≥ uε −εφuε|| ||K≥ uε ||K −ε φ|| || .

It follows that

1 ||≤ uε ||K≤ +1 2 |ε φ| || . (3.4)

In order to obtain a sequence ε →k 0 and a point u S∈ n−1 such that k

uε →u. If k→ ∞, it follows from (3.4) that || ||u K=_klim ||_→∞ uε_k || 1= . Namely, u

is a contain point of K and B2n. By (3.3), we obtain

1

|| || ( ), d ( ) ( ( )) , .

n x K− −p 〈∇h x x_K° φ 〉 γn x ≤ l Kp p〈u uφ 〉

∫

 

Taking

φ

_for −

φ

_{, we can find another contact point}u′_of_{K and} _B₂n such that

1

|| || ( ), d ( ) ( ( )) , .

n x K− −p 〈∇h x x_K° φ 〉 γn x ≥ l Kp p〈u u′φ ′〉

∫

 

Choosing φθ( )x = 〈x,θ θ〉 with θ∈Sn−1, we get (1.6). 

4. Estimate of the Distance

Lemma 4.1. (see [16]) Let x=( , , , )x x1 2  xn ∈n and

1 2

( , , , ) n n

y= y y  y ∈ . If

1 1 2 2

0<m ≤xk ≤M , 0<m ≤yk ≤M k, =1, , , n

then

2 1 2 1 2

2 2

1 1 2 1 .

n n n

k k k k

k k k

M M m m

m m M M

x y x y

= = =

 

+

 

  _≤   

   _ _  

   _ _  

 

 

∑

Lemma 4.1 implies that if _{x y}, _∈_n_{, then there exist a constant}_c_∈_(0,1) such that

| , |x y ≥c x y| || | . (4.1) Suppose that K is a centrally symmetric convex body in _n_{such that K is in}

the extremal position of (1.7). Now we estimate the geometric distance between K and B2n.

Theorem 4.1. Let K⊆B2n be a centrally symmetric convex body in n. If

K is in the extremal position of (1.7) and 1≤ <p 3, then

, 2n 2n,

n p

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where

1 2

,

1 2

( ) ( 1) _, _(0,1). 3

2 ( ) 2

p n

p

n p p

l B cp

c c

n p

π −

 

 ₊ 

 

= ∈

 _Γ − 

 

 

 

Proof. It follows from Corollary 3.2 that K satisfies

1 1

( )d ( ) , 1,

n

m m

n i i i i

i i

I x x µ x p c u u c

= =

=

_∫

⊗ −

∑

⊗

∑

=

 

d ( ) || || d ( ) / ( ( )) .p p

K n p

x x x l K

µ

_ ₌ −

γ



For y K_∈ °_and n 1

i

u S_∈ − _{. By (4.1), there exists a constant}_c_∈_(0,1)_such

that | ,〈y ui〉 ≥| c y| |. So we obtain

2 2 2 2

1

( | , | | | )d ( ) | | | | .

n

m i i

x y y µ x cp y c cp y

=

〈 〉 − ≥

∑

=

∫

 

That is,

2 2

(cp+1) | |y ≤

∫

_n| , | d ( ).〈x y〉 µ x

Since || || | , |x K≥ 〈x y〉 , we have

2

1

2 2

2 1

2 2

0 1

1 ₂

2 2

| , | || || d ( ) | , | d ( )

(2 ) | , | d ( ) d

3

2 ( ) | | .

2

n n

n

p p

K n n

n r

p n p

S p

p

x y x x x y x

y S r e r

p y

γ γ

π θ θ

π

−

− −

− ₋ ∞ _{− +} −

− − ₋

〈 〉 ≤ 〈 〉

= 〈 〉

−

= Γ

∫

 

From John’s theorem, for every centrally symmetric convex body K in _n_,

there is a corresponding to the ball λB2n such thatλB2n⊆K⊆ n Bλ 2n(λ>0).

Take λ =1/ n_{. We obtain} 1 _B₂n _{K B}₂n

n ⊆ ⊆ . Thus,

2 2

1 _{( )}n _{( )} _{( ).}n

p p p

l B l K l B

n ≤ ≤

Therefore, we get

1 1

2 2

3 2 ( )

2

| | ,

( ) ( 1)

p _p

n p

p n

y

l B π cp

−

 _Γ − 

 

≤

 + 

 

 



and the result yields.  Giannopoulos et al. in [5] proved that if K is in a position of maximal volume in L, then K L nK⊆ ⊆ _{, which is equivalent to}1 || || || || || ||x _K x _L x _K

n ≤ ≤ for all

n

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DOI: 10.4236/jamp.2018.612216 2598 Journal of Applied Mathematics and Physics ( )

1 .

( )

p

l L n l K

≤  ≤



Furthermore, let φ ∈GL( )n _{. Since} _φ_K_⊆_B₂n_{is in the maximal volume} posi-tion of K contained in B2n, we have 1 B2n K B2n

n ⊆φ ⊆ . Thus

2

( )

1 _1.

( )

p n p

l K

l B n

φ

≤  ≤



Finally, we propose the following concept of l0-norm: Let K be a convex

body in _n_{, we define}

0

l -norm by

0( ) exp( n || ||K n( )).

l K =

_∫

log x γ x



We propose an open question as follows: How should we solve the extreme problem

0

min{ (l φK o) : ∈φK⊆L,φ∈GL( )}?n

Funding

This work is supported by the National Natural Science Foundation of China (Grant No.11561020).

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this pa-per.

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