Journal of Function Spaces
Volume 2017 (2017), Article ID 2943073, 10 pages
https://doi.org/10.1155/2017/2943073

Research Article

The Characteristic Properties of the Minimal -Mean Width

Tongyi Ma

College of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, China

Correspondence should be addressed to Tongyi Ma; matongyi@126.com

Received 18 February 2017; Accepted 23 March 2017; Published 20 June 2017

Academic Editor: Antonio S. Granero

Copyright © 2017 Tongyi Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Giannopoulos proved that a smooth convex body has minimal mean width position if and only if the measure , supported on , is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal -mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal -mean width of convex bodies and prove the existence and uniqueness of the minimal -mean width in its images. In addition, we establish a characterization of the minimal -mean width, conclude the average with a variation of the minimal -mean width position, and give the condition for the minimum position of .

1. Introduction

Let denote the space of linear operators from to and . Suppose is a convex body in ; then is the family of its positions. In [1] it was shown that for many natural functionals of the form , , the solution of the problem is isotropic with respect to an appropriate measure depending on . The purpose of this note is to provide applications of this point of view in the case of the mean width functional under various constraints.

Recall that the width of in the direction of is defined by , where is the support function of convex . The mean width of is given by

where is the rotationally invariant probability measure on the unit sphere .

We say that has minimal mean width if for every . In [1], the authors show that a smooth enough convex body (that is, is twice continuously differentiable) is in minimal mean width position if and only if the measure , supported on , is isotropic.

Theorem 1. A smooth enough convex body in has minimal mean width if and only if

Moreover, if and has minimal mean width, we must have , where denotes rotation transformation group.

Yuan et al. in [2] introduced the notion of the minimal -mean width of convex body. Let be a convex body in and ; the -width of in the direction of is defined by . The -width function is translation invariant; therefore we may assume that . The -mean width of is given by

If for every , then we say that has minimal -mean width. The following isotropic characterization of the minimal -mean width position was proved in [2].

Theorem 2. A smooth enough convex body in has minimal -mean width if and only if

Moreover, if and has minimal -mean width, we must have .

It is easily seen that the -mean width belongs to the -Brunn-Minkowski theory. The -Brunn-Minkowski theory is far more general than the classical Brunn-Minkowski theory; we refer the reader to [3–25].

The aim of this article is to study the characterization of the minimal -mean width, which enriches the theory of -mean width of convex bodies. In Section 3, we discuss the continuity of the -mean width of convex bodies. In Section 4, we mainly demonstrate that the body has a unique image with minimal -mean width. In view of this fact, we define the minimal -mean width of by

In Section 5, we prove the following equivalent conditions with the minimal -mean width of convex bodies.

Theorem 3. Suppose is a smooth enough convex body in , , and . Then the following assertions are equivalent:

(1).(2)The measure is isotropic on .(3)For all , the transformation satisfies

In Section 6, we consider the average

of the norm on . Thus we define and obtain the following condition for the minimum position.

Theorem 4. Let be a symmetric convex body in and assume that for every . Then

In addition, we also get the following condition for the minimum position.

Theorem 5. Let be a symmetric convex body in satisfying and for every with . Then, for every we can find contact points , of and such that

where the condition means that but there exist contact points of and (see [1]).

Please see the next section for above interrelated notations, definitions, and their background materials.

2. Preliminaries

2.1. Notations

The setting will be the Euclidean -space . As usual, denotes the standard inner product of and in ; and denotes the unit ball and unit sphere in , respectively. The volume of is . We often use to denote the standard Euclidean norm, on occasion the total mass of a measure, and the absolute value of the determinant of an matrix. For brevity, we write , for .

Given a compact convex set in , its support function is defined by

The definition immediately gives that, for ,

As usual, write for the class of convex bodies in that contain the origin in their interiors. is often equipped with the Hausdorff metric , which is defined by

For , define by

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 and has continuous and positive radial function . Here, the radial function of , , is defined by

We write for the class of star bodies about the origin in . is often equipped with the dual Hausdorff metric , which is defined by

If convex body contains the origin as its interior point, then the Minkowski functional of is defined by

In this case, , where denotes the polar set of , which is defined by

If and any , then

In addition, we easily see that if , then the support and radial functions of , the polar body of , are defined, respectively, by

2.2. Linear Operators

For , and denote the transpose and norm of , respectively. For , let denote its maximal principal radius.

Two facts are in order (see [26]). First, is nondegenerated, if and only if the ellipsoid is nondegenerated. Second, for , since

it follows that

Let , for . Then the metric space is complete. Since is of finite dimension, a set in is compact, if and only if it is bounded and closed.

Lemma 6 (see [26]). Suppose . Then .

Thus, is bounded from above, if and only if is bounded from above.

Lemma 7 (see [26]). Suppose and with respect to . Then

(1) with respect to .(2) with respect to .(3) with respect to .

2.3. Notion of Isotropy of Measures

For further discussion, we introduce the important notion of isotropy of measures. A nonnegative Borel measure on is said to be isotropic if

Here, denotes the total mass of . The definition immediately yields

where denotes the th component of the coordinate of . For nonzero , the notation represents the linear operator of the rank on that takes to . It immediately gives the fact that . Equivalently, is isotropic if

where denotes the identity operator on . For more information about the isotropy, we refer to [1, 27–29].

The following fact will be needed.

Lemma 8 (see [30]). Suppose that is a probability measure on a space and is a -integrable function, where is a possibly infinite interval. Jensen’s inequality states that if is a convex function, then

If is strictly convex, equality holds if and only if is constant for , with almost all .

3. The Continuity of the -Mean Width

Using equivalent to uniformly on , together with the convexity of and definition (3), we immediately obtain the following.

Lemma 9. Suppose and , . If and , then

Fixing a convex body , the continuity of the -mean width with respect to is contained in the following.

Lemma 10. Suppose , , and . If , then

Proof. From (2) of Lemma 7, we have the following implications:

From the above, (35), and Lemma 9, we get .

Lemma 11. Suppose and . Then

Proof. Without loss of generality, we write in the form , where is an diagonal matrix, with and positive diagonal elements , and are orthogonal matrices. For any , let denote the -norm of , with . Recall that there exists a positive such that , and . Then from the definition of -mean width, Jensen’s inequality (Lemma 8, see [30, 31]), and (13), it follows that

It immediately yields

4. The Minimal -Mean Width

In order to demonstrate the existence and uniqueness of minimal -mean width, we begin by proving some lemmas.

Lemma 12. Suppose , , and . Then

Proof. From (3) and (11), we have

We conclude the proof.

For , from (18) and (19), we have

Therefore, we can get

Given an origin-symmetric ellipsoid in , let . Then there exists a such that, for all ,

Lemma 13. Suppose , , and . Then

Proof. From Lemma 12, (36), and Jensen’s inequality, together with (13), we have

We easily see that

Indeed, it is known that there exists a unit vector such that . Choose which is not orthogonal to . Then the statement follows by continuity. Accordingly, the desired inequality is derived.

With the previous lemmas in hand, we can show that the minimal -mean width is well-defined. Our purpose is to find necessary and sufficient conditions for a body to have minimal -mean width. We assume for simplicity that is twice continuously differentiable (we then say that is smooth enough).

Theorem 14. Suppose and any . Then, modulo orthogonal transformations, there exists a unique solution to the minimization problem

Proof. Note that the proof is based on the idea of Zou and Xiong [26, 32]. Let be a minimizing sequence for the problem; that is,

Note that

which implies ; therefore is bounded from above. So, by Lemma 13, is bounded with respect to the Hausdorff metric. From the Blaschke selection theorem, has a convergent subsequence that converges to a body . Since volume functional is continuous with respect to the Hausdorff metric, and for each , it yields that ; since the convergence of is equivalent to the uniform convergence of on , and for all , it yields that for all . Thus, is a nondegenerated origin-symmetric ellipsoid.
Consequently, there exists a transformation such that . This demonstrates the existence of solutions to the considered problem.
Now, we prove the uniqueness by contradiction. Assume that both solve the considered minimization problem. Let and . It is known that each can be represented in the form , where is symmetric, positive definite and is orthogonal. So, without loss of generality, we may assume that and are symmetric and positive definite.
The Minkowski inequality for symmetric positive definite matrices shows that

Let

Then , and, for all ,

Using (32) and (45) and noticing that if and , then (where if , and if ), we have

Namely,

However, by the previous assumption on and , we have

which is a contradiction. The proof is complete.

In view of Theorem 14, naturally, we introduce the following.

Definition 15. Suppose and . The quantity

is called the minimal -mean width of the convex body with respect to .

5. The Characterization of the Minimal -Mean Width

Theorem of this section characterizes the convex body with minimal -mean width.

Theorem 16. Suppose , , and . Then the following assertions are equivalent:

(1).(2)The measure is isotropic on .(3)For all , the transformation satisfies

Proof. First, we prove the equivalence of (1) and (2). Suppose that (1) holds. Since is invariant, we may assume that is the identity matrix .
Let be a linear transformation. Then there exists such that, for each , the matrices are still positive definite. For , define

Then is volume preserving. If is sufficiently small, from (33), together with the two equalities (see [1])

and the definition of , we have

From the smoothness of in , the integrand depends smoothly on . Thus,

Calculating it directly, we have

Let and . Using the facts and , it follows that

where . Thus, is isotropic on .
Next, we show the implication “”. The proof will be completed by two steps.
Firstly, for a point , let

where denotes the diagonal matrix with diagonal elements .
We aim to show that

Here, denotes the point .
It can be checked that is continuous and convex and is strictly increasing in , for any . Thus, is compact, convex, and of nonempty interior. Namely, it is a convex body.
By using the fact that is smooth in uniformly for , we have

Meanwhile, since the boundary of is given by the equation with , so the vector is an outer normal of at the boundary point . Notice that is isotropic; it yields

Thus, is an outer normal vector of at the boundary point . Consequently,

That is to say, for all , if , then . In contrast, for all with , the arithmetic-geometric mean inequality yields that , with equality if and only if . Thus, (58) is derived.
Secondly, with (58) in hand, we aim to show that, for all , , with equality if and only if is orthogonal.
It is known that can be represented in the form , where and are orthogonal matrices, and is diagonal and positive definite. Therefore, by (33) and (58), we have

Equality holds if and only if , equivalently, if and only if is orthogonal. Thus, the implication “” is shown.
In the remaining part, we prove the equivalence of (2) and (3). From the definitions of and (11), we have

which immediately yields that

Meanwhile, for , we have

Therefore,

With these, the equivalence of (2) and (3) is shown. The proof is complete.

6. The Characterization of the Average

In this section, we consider the average

of the norm on and define .

We conclude this section with a variation of the minimal -mean width position: consider a symmetric convex body in and the problem of minimizing over all .

The following fact will be needed.

Lemma 17 (see [2]). Let be a smooth enough convex body in . We define

Then

for every .

Now, we obtain the following condition for the minimum position.

Theorem 18. Let be a symmetric convex body in and assume that for every . Then

for every .

Proof. Without loss of generality, we may assume that is smooth enough. Let and be small enough, and write . Then (see [1])

Our assumption about takes the form

Note that there are two equalities (see [33])

which implies

Letting and replacing by , we have

for every . Using (75) with , , we get

for every . Taking into account Lemma 17, we conclude the proof.

When , it is easy to show that Theorem 18 reduces to Giannopoulos and Milman’s result (see [1]).

Recall that the condition means that but there exist contact points of and (see [1]). We also obtain the following result.

Theorem 19. Let be a symmetric convex body in satisfying and for every with . Then, for every we can find contact points , of and such that

Proof. Let and be small enough. Then

satisfies . Therefore

Notice that (see [33])

It follows from (79) that

Let be a point on at which the minimum is attained. Since and , and noticing that

(81) takes the form

If is a contact point of and , we must have

It follows that

Also, and using (85) we get . That is, is a contact point of and . Now, we can find a sequence and a point such that . Letting in (83), we obtain

Replacing by we find another contact point of and such that

Choosing , , and applying Lemma 17, we obtain (77).

When , it is easy to show that Theorem 19 reduces to Giannopoulos and Milman’s result (see [1]).

Conflicts of Interest

The author declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundations of China (Grant no. 11561020) and is partly supported by the National Natural Science Foundations of China (Grant no. 11371224).

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