Journal of Applied Mathematics and Physics
Vol.06 No.12(2018), Article ID:89414,11 pages
10.4236/jamp.2018.612216
Extremal Problems Related to Dual Gauss-John Position
Tongyi Ma
College of Mathematics and Statistics, Hexi University, Zhangye, China
Received: December 4, 2018; Accepted: December 23, 2018; Published: December 26, 2018
ABSTRACT
In this paper, the extremal problem, , of two convex bodies K and L in 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: . As an application, we give the geometric distance between the unit ball and a centrally symmetric convex body K.
Keywords:
Dual Gauss-John Position, Optimization Theorem of John, Dual -Norm, Contact Pair
1. Introduction
Let be the classical Gaussian probability measure with density ,
and is the Minkowski functional of a convex body . An important quantity on local theory of Banach space is the associated l-norm:
The minimum of the functional
under the constraint is attained for , then a convex body K is in the Gauss-John position, where , is the Euclidean unit ball and is the identity mapping from to .
For , the map is the rank 1 linear operator .
Giannopoulos et al. in [1] showed that if K is in the Gauss-John position, then there exist contact points , and constants such that and
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 and the John ellipsoid is of order .
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 is the maximal ellipsoid in K, it can be characterized by points of contact between the boundary of K and that of . 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 is a position of K if , for some non-degenerate linear mapping and some . We say that K is in a position of maximal volume in L if and for any position of K such that we have , where denotes the volume of appropriate dimension.
Recently, Li and Leng in [14] generalized the Gauss-John position to a general situation. For , denote -norm by
(1.1)
They consider the following extremal problem:
(1.2)
where L is a given convex body in and K is a convex body containing the origin o such that .
Li and Leng [14] showed that let L be a given convex body in and K be a convex body such that . If K is in extremal position of (1.2), then there exist contact pairs of , and constants such that
where is the probability measure on with normalized density
In this paper, we first present a dual concept of -norm . 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 , we define the dual -norm of convex body K by
(1.3)
where is the radial function of the star body K about the origin.
Now, we consider the extremal problem:
(1.4)
where L is a given convex body in and K is a convex body containing the origin o such that .
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 and K be a convex body such that . If K is in extremal position of (1.4), then there exist contact pairs of , and such that
where is the probability measure on with normalized density
Next the following result is obtained, which is an restriction that is weaker than the extremal problem (1.4):
(1.5)
Theoren 1.2. Let K be a given convex body in . If is the solution of the extremal problem (1.5), then there exist contact points of K and such that
(1.6)
for every .
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
(1.7)
Section 3 shows the geometric distance between the unit ball 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 equipped with the canonical Euclidean scalar product and write for the corresponding Euclidean norm. We denote the unit sphere by .
Let K be a convex body (compact, convex sets with non-empty interiors) in . The support function of K is defined by
Obviously, for , where denotes the transpose of .
A set is said to be a star body about the origin, if the line segment from the origin to any point is contained in K and K has continuous and positive radial function . Here, the radial function of , is defined by
Note that if K be a star body (about the origin) in , then K can be uniquely determined by its radial function and vice verse. If , we have
and
More generally, from the definition of the radial function it follows immediately that for the radial function of the image of star body K is given by , for all .
If and (not both zero), then for , the -radial combination, , is defined by (see [15])
(2.1)
If a star body K contains the origin o as its interior point, then the Minkowski functional of K is defined by
In this case,
where denotes the polar set of K, which is defined by
It is easy to verify that for ,
where denotes the reverse of the transpose of . Obviously, (see [13] for details).
Let K and L be two convex bodies in . According to [4], if , we call a pair a contact pair for if it satisfies:
1) ,
2) ,
3) .
If , we denote by the rank one projection defined by for all .
The geometric distance of the convex bodies K and L is defined by
3. Proof of Main Results
First, we prove that is a norm with respect to -radial combination in . Apparently, and if and only if . At the same time, if real constant . In addition, it is follows that
Indeed, we have
Therefore,
is a norm with respect to
-radial combination and
is normed space for
.
Now, we prove the optimization theorem of John [2] (see [10] also).
Lemma 3.1. Let be a -function. Let S be a compact metric space and be continuous. Suppose that for every , exists and is continuous on .
Let and satisfy
Then, either , or, for some , there exist and such that for , and
Using a similar argument as that in [1], we give the proof of Theorem 1.1.
Proof of Theorem 1.1. For , we define by
(3.1)
where is the linear mapping from to . Clearly is . For , define by
The set
is just the set of elements such that . If K is in extremal position of , then attains its minimum on at , namely,
Now we prove . It follows from (3.1) that
It is easy to obtain that for non-degenerate , we have
and
where , denotes conjugate of transposed transformation of , and is inverse transform of .
Since attains its minimum on at , combining with Lemma 3.1, it follows that for some , there exist , , , , such that
and
(3.2)
From , we yield and . Taking the trace in (3.2), we have
Suppose . Together with (3.2), we obtain
where . This completes the proof.
If and , 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 . If K is in extremal position of (1.7), then there exist contact points with and , such that,
where is the probability measure on with normalized density
Proof of Theorem 1.2. Suppose that and is small enough. Then
satisfies . Therefore
Let be a point on at which the minimum is attained. Observe that
and
Since and , we have
(3.3)
If u is a contact point of K and , then
It follows that
(3.4)
In order to obtain a sequence and a point such that . If , it follows from (3.4) that . Namely, u is a contain point of K and . By (3.3), we obtain
Taking for , we can find another contact point of K and such that
Choosing with , we get (1.6).
4. Estimate of the Distance
Lemma 4.1. (see [16]) Let and . If
then
Lemma 4.1 implies that if , then there exist a constant such that
(4.1)
Suppose that K is a centrally symmetric convex body in such that K is in the extremal position of (1.7). Now we estimate the geometric distance between K and .
Theorem 4.1. Let be a centrally symmetric convex body in . If K is in the extremal position of (1.7) and , then
where
Proof. It follows from Corollary 3.2 that K satisfies
where is the probability measure on with normalized density
For and . By (4.1), there exists a constant such that . So we obtain
That is,
Since , we have
From John’s theorem, for every centrally symmetric convex body K in , there is a corresponding to the ball such that . Take . We obtain . Thus,
Therefore, we get
and the result yields.
Giannopoulos et al. in [5] proved that if K is in a position of maximal volume in L, then , which is equivalent to for all . Hence it follows that
Furthermore, let . Since is in the maximal volume position of K contained in , we have . Thus
Finally, we propose the following concept of -norm: Let K be a convex body in , we define -norm by
We propose an open question as follows: How should we solve the extreme problem
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 paper.
Cite this paper
Ma, T.Y. (2018) Extremal 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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