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 JAMP  Vol.6 No.12 , December 2018
Extremal Problems Related to Dual Gauss-John Position
Abstract: In this paper, the extremal problem, min, of two convex bodies K and L in ℝn is considered. For K to be in extremal position in terms of a decomposition of the identity, give necessary conditions together with the optimization theorem of John. Besides, we also consider the weaker optimization problem: min. As an application, we give the geometric distance between the unit ball B2n and a centrally symmetric convex body K.

1. Introduction

Let γ n be the classical Gaussian probability measure with density 1 ( 2 π ) n e | x | 2 2 ,

and | | | | K is the Minkowski functional of a convex body K n . An important quantity on local theory of Banach space is the associated l-norm:

l ( K ) = n | | x | | K d γ n ( x ) .

The minimum of the functional

n x ϕ K d γ n (x)

under the constraint ϕ K B 2 n is attained for ϕ = I n , then a convex body K is in the Gauss-John position, where ϕ GL ( n ) , B 2 n is the Euclidean unit ball and I n 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 x 1 , x 2 , , x m K S n 1 , and constants c 1 , c 2 , , c m > 0 such that i = 1 m c i = 1 and

n ( x x I n ) | | x | | K d γ n ( x ) = n | | x | | K d γ n ( x ) ( i = 1 m c i x i x i ) .

Note that the Gauss-John position is not equivalent to the classical John position. Giannopoulos et al. [1] pointed out that, when K is in the Gauss-John position, the distance between the unit ball B 2 n and the John ellipsoid is of order n / log 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 B 2 n is the maximal ellipsoid in K, it can be characterized by points of contact between the boundary of K and that of B 2 n . 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 information 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 ˜ = ϕ K + 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 n ( K ˜ ) vol n ( K ) , where vol n ( ) denotes the volume 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

l p ( K ) = ( n | | x | | K p d γ n ( x ) ) 1 p . (1.1)

They consider the following extremal problem:

min { l p ( ϕ K ) : o ϕ K 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 ( x i , y i ) 1 i m of ( K , L ) , and constants c 1 , c 2 , , , c m > 0 such that

I n = n ( x x ) d μ ( x ) p i = 1 m c i x i y i , i = 1 m c i = 1 ,

where d μ ( x ) is the probability measure on n with normalized density

d μ ( x ) = | | x | | K p d γ n ( x ) / ( l p ( K ) ) p .

In this paper, we first present a dual concept of l p -norm l p ( K ) . The generalizations 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

l ˜ p ( K ) = ( n ρ K ( x ) p d γ n ( x ) ) 1 p , (1.3)

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

Now, we consider the extremal problem:

min { l ˜ p ( ϕ K ) : o ϕ K 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 m n 2 contact pairs ( x i , y i ) 1 i m of ( K , L ) , and c 1 , c 2 , , c m > 0 such that

I n = n ( x x ) d μ ˜ ( x ) p i = 1 m c i x i y i , i = 1 m c i = 1 ,

where d μ ˜ ( x ) is the probability measure on n with normalized density

d μ ˜ ( x ) = | | x | | K p d γ n ( x ) / ( l ˜ p ( K ) ) p .

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

min { ( l ˜ p ( ϕ K ) ) p : ϕ K B 2 n , ϕ K S n 1 , ϕ GL ( n ) } . (1.5)

Theoren 1.2. Let K be a given convex body in n . If I n is the solution of the extremal problem (1.5), then there exist contact points u , u of K and B 2 n such that

u , θ 2 ( l ˜ p ( K ) ) p n | | x | | K p 1 h K o ( x ) , θ x , θ d γ n ( x ) u , θ 2 , (1.6)

for every θ S n 1 .

The rest of this paper is organized as follows: In Section 2, some basic notation 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

min { ( l ˜ p ( ϕ K ) ) p : o ϕ K B 2 n , ϕ GL ( n ) } , (1.7)

Section 3 shows the geometric distance between the unit ball B 2 n and a centrally symmetric convex body K.

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 Euclidean scalar product , and write | | for the corresponding Euclidean norm. We denote the unit sphere by S n 1 .

Let K be a convex body (compact, convex sets with non-empty interiors) in n . The support function of K is defined by

h K ( x ) = max { x , y : y K } , x n .

Obviously, h ϕ K ( x ) = h K ( ϕ t x ) for ϕ GL ( n ) , where ϕ t denotes the transpose 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 , ρ K : S n 1 [ 0 , ) , is defined by

ρ K ( u ) = max { λ : λ u K } .

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

ρ K ( α x ) = α 1 ρ K ( x ) and ρ α K ( x ) = α ρ K ( x ) .

More generally, from the definition of the radial function it follows immediately that for ϕ GL ( n ) the radial function of the image ϕ K = { ϕ y : y K } of star body K is given by ρ ϕ K ( x ) = ρ K ( ϕ 1 x ) , for all x n .

If K , L S o n and λ , μ 0 (not both zero), then for p > 0 , the L p -radial combination, λ K + ˜ p μ L S o n , is defined by (see [15])

ρ ( λ K + ˜ p μ L , ) p = λ ρ ( K , ) p + μ ρ ( L , ) p . (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 λ K } .

In this case,

| | x | | K = ρ K 1 ( x ) = h K ° ( x ) ,

where K ° denotes the polar set of K, which is defined by

K ° = { x n : x , y 1 forall 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 o K L n , we call a pair ( x , y ) n × n a contact pair for ( K , L ) if it satisfies:

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 δ G ( K , L ) of the convex bodies K and L is defined by

δ G ( K , L ) = inf { α β : α > 0 , β > 0 , ( 1 / β ) L K α L } .

3. Proof of Main Results

First, we prove that l ˜ p ( ) is a norm with respect to L p -radial combination in S o n . Apparently, l ˜ p ( K ) 0 and l ˜ p ( K ) = 0 if and only if K = { o } . At the same time, l ˜ p ( c K ) = c l ˜ p ( K ) if real constant c > 0 . In addition, it is follows that

l ˜ p ( K + ˜ p L ) l ˜ p ( K ) + l ˜ p ( L ) .

Indeed, we have

l ˜ p ( K + ˜ p L ) = ( n ρ K + ˜ p L p ( x ) d γ n ( x ) ) 1 p = ( n ρ K p ( x ) d γ n ( x ) + n ρ L p ( x ) d γ n ( x ) ) 1 p ( n ρ K p ( x ) d γ n ( x ) ) 1 p + ( n ρ L p ( x ) d γ n ( x ) ) 1 p = l ˜ p ( K ) + l ˜ p ( L ) .

Therefore, l ˜ p ( ) is a norm with respect to L p -radial combination and S o n is normed space for
l ˜ p ( ) .

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

Lemma 3.1. Let F : N be a C 1 -function. Let S be a compact metric space and G : N × S be continuous. Suppose that for every s S , z G ( z , s ) exists and is continuous on N × S .

Let A = { z N : G ( z , s ) 0 , forall s S } and z 0 A satisfy

F ( z 0 ) = min z A F ( z ) .

Then, either z F ( z 0 ) = 0 , or, for some 1 m N , there exist s 1 , s 2 , , s m S and λ 1 , λ 2 , , λ m such that G ( z 0 , s i ) = 0 , λ i 0 for 1 i m , and

z F ( z 0 ) = i = 1 m λ i z G ( z 0 , s i ) .

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 F : N by

F ( ϕ ) = l ˜ p ( ϕ K ) = ( n | | ϕ 1 x | | K p d γ n ( x ) ) 1 p , (3.1)

where ϕ N is the linear mapping from n to n . Clearly F is C 1 . For S = K × L ° , define G : N × S by

G ( ϕ , ( x , y ) ) = 1 ϕ x , y .

The set

A = { z N : G ( z , s ) 0 , s S }

is just the set of elements ϕ N such that ϕ K L . If K is in extremal position of min { l ˜ p ( ϕ K ) : o ϕ K L , ϕ GL ( n ) } , then F attains its minimum on A at I n , namely,

F ( I n ) = l ˜ p ( K ) = min { l ˜ p ( ϕ K ) : o ϕ K L , ϕ GL ( n ) } .

Now we prove ϕ F ( I n ) . It follows from (3.1) that

F ( ϕ ) = ( n | | ϕ 1 x | | K p d γ n ( x ) ) 1 p = ( ( 2 π ) n 2 n | | ϕ 1 x | | K p e | x | 2 2 d x ) 1 p = ( ( 2 π ) n 2 ( det ϕ ) n | | x | | K p e | ϕ x | 2 2 d x ) 1 p .

It is easy to obtain that for non-degenerate ϕ , we have

ϕ G ( ϕ , ( x , y ) ) = ϕ ϕ x , y = ϕ x y , ϕ = x y

and

ϕ F ( ϕ ) = 1 p ( ( 2 π ) n 2 ( det ϕ ) n | | x | | K p e | ϕ x | 2 x d x ) 1 q × [ ( 2 π ) n 2 ( det ϕ ) ( ϕ 1 ) n | | x | | K p e | ϕ x | 2 x d x ( 2 π ) n 2 ( det ϕ ) n | | x | | K p e | ϕ x | 2 x x x d x ] ,

where 1 p + 1 q = 1 , ( ϕ 1 ) * denotes conjugate of transposed transformation of ϕ 1 , and ϕ 1 is inverse transform of ϕ GL ( n ) .

Since F attains its minimum on A at z 0 = I n , combining with Lemma 3.1, it follows that for some m N , there exist λ i 0 , s i S , s i = ( x i , y i ) , 1 i m , such that

x i , y i = 1 G ( I n , ( x i , y i ) ) = 1 , 1 i m ,

and

ϕ F ( I n ) = 1 p ( l ˜ p ( K ) ) p q n ( I n x x ) | | x | | K p d γ n ( x ) = i = 1 m λ i ϕ G ( I n , ( x i , y i ) ) = i = 1 m λ i x i y i . (3.2)

From x i , y i = 1 , x i K L , y i L K , we yield x i L and y i K . Taking the trace in (3.2), we have

Tr ( ϕ F ( I n ) ) = Tr ( 1 p ( l ˜ p ( K ) ) p q n ( I n x x ) | | x | | K p d γ n ( x ) ) = 1 p ( l ˜ p ( K ) ) p q [ n n | | x | | K p d γ n ( x ) n | x | 2 | | x | | K p d γ n ( x ) ] = 1 p ( l ˜ p ( K ) ) p q [ n 0 r n p 1 e r 2 2 d r 0 r n p + 1 e r 2 2 d r ] S n 1 | | θ | | K p d S ( θ ) = 1 p ( l ˜ p ( K ) ) p q ( p n | | x | | K p d γ n ( x ) ) = l ˜ p ( K ) .

Suppose λ i = c i l ˜ p ( K ) . Together with (3.2), we obtain

n ( x x I n ) | | x | | K p d γ n ( x ) = p ( l ˜ p ( K ) ) p ( i = 1 m c i x i y i ) ,

where i = 1 m c i = 1 . This completes the proof.

If L = B 2 n and G ( ϕ , x ) = 1 | ϕ x | 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 B 2 n . If K is in extremal position of (1.7), then there exist contact points u 1 , u 2 , , u m K S n 1 with m n 2 and c 1 , c 2 , , c m > 0 , such that,

I n = n ( x x ) d μ ˜ ( x ) p i = 1 m c i u i u i , i = 1 m c i = 1 ,

where d μ ˜ ( x ) is the probability measure on n with normalized density

d μ ˜ ( x ) = | | x | | K p d γ n ( x ) / ( l ˜ p ( K ) ) p .

Proof of Theorem 1.2. Suppose that ϕ L ( n , n ) and ε > 0 is small enough. Then

ϕ 1 : = ( min u S n 1 | | u ε ϕ u | | K ) ( I n ε ϕ ) 1

satisfies ϕ 1 K B 2 n , ϕ 1 K S n 1 . Therefore

n | | x ε ϕ x | | K p d γ n ( x ) ( l ˜ p ( K ) ) p ( min u S n 1 | | u ε ϕ u | | K ) p .

Let u ε be a point on S n 1 at which the minimum is attained. Observe that

| | x ε ϕ x | | K p = | | x | | K p + ε p | | x | | K p 1 h K ° ( x ) , ϕ x + O (ε2)

and

| u ε ε ϕ u ε | p = 1 + ε p u ε , ϕ u ε + O ( ε 2 ) .

Since u ε S n 1 and | | | | K | | , we have

n p | | x | | K p 1 h K ( x ) , ϕ x d γ n ( x ) + O ( ε ) ( l ˜ p ( K ) ) p ( min u S n 1 | | u ε ϕ u | | K ) p 1 ε ( l ˜ p ( K ) ) p | u ε ε ϕ u ε | p 1 ε = ( l ˜ p ( K ) ) p ( p u ε , ϕ u ε + O ( ε ) ) . (3.3)

If u is a contact point of K and B 2 n , then

1 + ε | | ϕ | | | | u ε ϕ u | | K | | u ε ε ϕ u ε | | K | | u ε | | K ε | | ϕ | | .

It follows that

1 | | u ε | | 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 u ε k u . If k , it follows from (3.4) that | | u | | K = lim k | | u ε k | | = 1 . Namely, u is a contain point of K and B 2 n . By (3.3), we obtain

n | | x | | K p 1 h K ° ( x ) , ϕ x d γ n ( x ) ( l ˜ p ( K ) ) p u , ϕ u .

Taking ϕ for ϕ , we can find another contact point u of K and B 2 n such that

n | | x | | K p 1 h K ° ( x ) , ϕ x d γ n ( x ) ( l ˜ p ( K ) ) p u , ϕ u .

Choosing ϕ θ ( x ) = x , θ θ with θ S n 1 , we get (1.6).

4. Estimate of the Distance

Lemma 4.1. (see [16]) Let x = ( x 1 , x 2 , , x n ) n and y = ( y 1 , y 2 , , y n ) n . If

0 < m 1 x k M 1 , 0 < m 2 y k M 2 , k = 1 , , n ,

then

( k = 1 n x k 2 ) ( k = 1 n y k 2 ) ( M 1 M 2 m 1 m 2 + m 1 m 2 M 1 M 2 2 ) 2 ( k = 1 n x k y k ) 2 .

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 B 2 n .

Theorem 4.1. Let K B 2 n be a centrally symmetric convex body in n . If K is in the extremal position of (1.7) and 1 p < 3 , then

c ˜ n , p B 2 n K B 2 n ,

where

c ˜ n , p = l ˜ p ( B 2 n ) n ( π ( c p + 1 ) 2 1 p 2 Γ ( 3 p 2 ) ) 1 p , c ( 0 , 1 ) .

Proof. It follows from Corollary 3.2 that K satisfies

I n = n ( x x ) d μ ˜ ( x ) p i = 1 m c i u i u i , i = 1 m c i = 1 ,

where d μ ˜ ( x ) is the probability measure on n with normalized density

d μ ˜ ( x ) = | | x | | K p d γ n ( x ) / ( l ˜ p ( K ) ) p .

For y K ° and u i S n 1 . By (4.1), there exists a constant c ( 0 , 1 ) such that | y , u i | c | y | . So we obtain

n ( | x , y | 2 | y | 2 ) d μ ˜ ( x ) c p | y | 2 i = 1 m c i = c p | y | 2 .

That is,

( c p + 1 ) | y | 2 n | x , y | 2 d μ ˜ ( x ) .

Since | | x | | K | x , y | , we have

n | x , y | 2 | | x | | K p d γ n ( x ) n | x , y | 2 p d γ n ( x ) = ( 2 π ) n 2 S n 1 | θ , y | 2 p d S ( θ ) 0 r n p + 1 e r 2 2 d r = 2 1 p 2 π 1 2 Γ ( 3 p 2 ) | y | 2 p .

From John’s theorem, for every centrally symmetric convex body K in n , there is a corresponding to the ball λ B 2 n such that λ B 2 n K n λ B 2 n ( λ > 0 ) . Take λ = 1 / n . We obtain 1 n B 2 n K B 2 n . Thus,

1 n l ˜ p ( B 2 n ) l ˜ p ( K ) l ˜ p ( B 2 n ) .

Therefore, we get

| y | n l ˜ p ( B 2 n ) ( 2 1 p 2 Γ ( 3 p 2 ) π ( c p + 1 ) ) 1 p ,

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 n K , which is equivalent to 1 n | | x | | K | | x | | L | | x | | K for all x n . Hence it follows that

1 l ˜ p ( L ) l ˜ p ( K ) n .

Furthermore, let ϕ GL ( n ) . Since ϕ K B 2 n is in the maximal volume position of K contained in B 2 n , we have 1 n B 2 n ϕ K B 2 n . Thus

1 n l ˜ p ( ϕ K ) l ˜ p ( B 2 n ) 1.

Finally, we propose the following concept of l 0 -norm: Let K be a convex body in n , we define l 0 -norm by

l 0 ( K ) = exp ( n l o g | | x | | K γ n ( x ) ) .

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

min { l 0 ( ϕ K ) : o ϕ K L , ϕ GL ( n ) } ?

Funding

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

Cite this paper: Ma, T. (2018) Extremal Problems Related to Dual Gauss-John Position. Journal of Applied Mathematics and Physics, 6, 2589-2599. doi: 10.4236/jamp.2018.612216.
References

[1]   Giannopoulos, A., Milman, V. and Rudelson, M. (2000) Convex Bodies with Minimal Mean Width. Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, 1745, Springer, Berlin, 81-93.

[2]   John, F. (1948) Extremum Problems with Inequalities as Subsidiary Conditions. Studies and Essays Presented to R. Courant on His 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 187-204.

[3]   Lewis, D. (1979) Ellipsoids Defined by Banach Ideal Norms. Mathematika, 26, 18-29. https://doi.org/10.1112/S0025579300009566

[4]   Tomczak-Jaegermann, N. (1989) Banach-Mazur Distances and Finite-Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, 38, Pitman, London.

[5]   Giannopoulos, A., Perissinaki, I. and Tsolomitis, A. (2001) Johns Theorem for an Arbitrary Pair of Convexbodies. Geom. Dedicata, 84, 63-79. https://doi.org/10.1023/A:1010327006555

[6]   Bastero and Romance, M. (2002) Johns Decomposition of the Identity in the Non-Convex Case. Positivity, 6, 1-16. https://doi.org/10.1023/A:1012087231191

[7]   Ball, K. (1997) An Elementary Introduction to Modern Convex Geometry. In: Flavors of Geometry, Math. Sci.Res. Inst. Publ., 31, Cambridge University Press, Cambridge.

[8]   Bastero, J., Bernues, J. and Romance, M. (2007) From John to Gauss-John Positions via Dual Mixed Volumes. J. Math. Anal. Appl., 328, 550-566. https://doi.org/10.1016/j.jmaa.2006.05.047

[9]   Giannopoulos, A. and Milman, V.D. (2000) Extremal Problems and Isotropic Positions of Convex Bodies. Israel J. Math., 117, 29-60. https://doi.org/10.1007/BF02773562

[10]   Gordon, Y., Litvak, A.E., Meyer, M., et al. (2004) Johns Decomposition in the General Case and Applications. J. Differential Geom., 68, 99-119. https://doi.org/10.4310/jdg/1102536711

[11]   Li, A.J., Wang, G. and Leng, G. (2011) An Extended Loomis-Whitney Inequality for Positive Double John Bases. Glasg. Math. J., 53, 451-462. https://doi.org/10.1017/S0017089511000061

[12]   Li, A.J. and Leng, G. (2011) Brascamp-Lieb Inequality for Positive Double John Basis and Its Reverse. Sci. China Math., 54, 399-410. https://doi.org/10.1007/s11425-010-4093-5

[13]   Pisier, G. (1989) The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, 94, Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511662454

[14]   Li, A.J. and Leng, G.S. (2012) Extremal Problems Related to Gauss-John Position. Acta Mathematica Sinica, English Series, 28, 2527-2534. https://doi.org/10.1007/s10114-012-9735-9

[15]   Gardner, R.J. (2002) The Brunn-Minkowski Inequality. Bull. Amer. Math. Soc., 39, 355-405. https://doi.org/10.1090/S0273-0979-02-00941-2

[16]   Pólya, G. and Szegö, G. (1925) Aufgaben und Lehrsãtzeaus der Analysis. Vol. 1, Berlin, p. 57, 213-214.

 
 
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