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Estimates for the extremal sections of complex -balls
Journal of Inequalities and Applications volume 2014, Article number: 235 (2014)
The problem of maximal hyperplane section of with is considered, which is the complex version of central hyperplane section problem of . The relation between the complex slicing problem and the complex isotropic constant of a body is established, an upper bound estimate for the volume of complex central hyperplane sections of normalized complex -balls that does not depend on n and p is shown, which extends results of Oleszkiewicz and Pełczyński, Koldobsky and Zymonopoulou, and Meyer and Pajor.
Let and denote the unit balls of the real and complex n-dimensional spaces, and , respectively. We denote by the n-dimensional Lebesgue measure of a compact set K. We write and such that and , respectively.
The extremal volume of central hyperplane section of is studied by various authors (see, e.g., [1–8]). Especially, the left-hand side of the following inequalities is known due to Meyer and Pajor  for all and , and Schmuckenschläger  for all . The right-hand side of the following inequalities is due to Ma and the third named author , and it shows an upper bound estimate for the volume of central hyperplane sections of normalized -balls that does not depend on n and p.
Theorem 1.1 Let , , , and H any central hyperplane in . Then
Moreover, the minimum occurs for if ξ has only one non-zero coordinate where ξ is the normal vector of H.
Note that Theorem 1.1 is proved by determining the extremal value of the isotropic constant of together with the well-known relation between the slicing problem and the isotropic constant of a body. Motivated by this idea, we define a new quantity, called the complex isotropic constant, and establish its relation to the complex slicing problem. Thus, the complex version of Theorem 1.1 (see Theorem 1.2) can be proved by estimating the extremal value of the complex isotropic constant of .
A noteworthy fact is that the extremal volume of complex central hyperplane section of has not been studied until recent years. Other results concerning convex bodies in a complex vector space as ambient space can be found in [9–19]. Especially, in , Oleszkiewicz and Pełczyński proved that , where is the complex central hyperplane (see Section 2.1 for the definition). Furthermore, they showed that the minimal sections are the ones orthogonal to vectors with only one non-zero coordinate, and the maximal sections are orthogonal to vectors of the form , where , and are standard basic vectors, and , . In , Koldobsky and Zymonopoulou studied the extremal sections of , for and showed that the minimum corresponds to hyperplanes orthogonal to vectors with and the maximum corresponds to hyperplanes orthogonal to vectors with only one non-zero coordinate. Moreover, a result of Meyer and Pajor [, Corollary 2.5] states the following. Suppose , then ; Suppose , then , where . Recently, Koldobsky and König  considered minimal volume of slabs for the complex cube.
The case of the following theorem follows directly from the work of Koldobsky and Zymonopoulou . The following theorem extends their results to , and it also shows an upper bound estimate for the volume of complex central hyperplane sections of normalized complex -balls that does not depend on n and p.
Theorem 1.2 Let , , , and any complex hyperplane in . Then
Moreover, the minimum occurs for if ξ has only one non-zero coordinate.
Our problem is different from the extremal volume of central hyperplane section of problem in two aspects. First, except ; Secondly, we do only -dimensional sections, sections by subspaces coming from complex hyperplanes, rather than all -dimensional sections, and -dimensional sections in real case.
2 Notations and preliminaries
2.1 Background on complex vector space
Throughout this paper, we denote their real scalar product by and the Euclidean norm of x by for . For , their complex scalar product by and the modulus of x by .
Origin-symmetric convex bodies in are the unit balls of norms on . We denote by the norm corresponding to the body K:
We identify with using the standard mapping, that is, for
Since norms on satisfy the equality
origin-symmetric complex convex bodies correspond to those origin-symmetric convex bodies K in that are invariant with respect to any coordinate-wise two-dimensional rotation, namely for each and each
where stands for the counterclockwise rotation of by the angle θ with respect to the origin. We define the map :
as for each .
For , , denote by
the complex hyperplane through the origin, perpendicular to ξ. Under the standard mapping from to the hyperplane turns into a -dimensional subspace of orthogonal to the vectors
The orthogonal two-dimensional subspace has an orthonormal basis ξ, .
Let be the -balls, when viewed as a subset of :
if , and
If , is -invariant convex body in . Here a convex body is a compact convex set with non-empty interior.
2.2 Complex isotropic bodies
First, noting that a subset is called a complex convex body means that K is a convex body in under the map (2.1). An important notion in asymptotic convex geometry is the quantity called isotropic constant (see, e.g., [20–28]).
Recall that a convex body K in is called isotropic with the isotropic constant if , the origin is the center of mass of K, and
for every .
Definition A body is called complex isotropic with the complex isotropic constant , if , the origin is the center of mass of K, and
for all .
This definition is natural since we identify with using the mapping τ and
where denotes the identity operator on , and is the rank 1 linear operator on that takes y to . More precisely, (2.6) means
Summing (2.9) with , we have
Now, we show the relation between real isotropic bodies and complex isotropic bodies when we consider the convex body defined in .
Theorem 2.1 Let be a convex body with , center of mass at the origin. Then
if K is (real) isotropic, then K is complex isotropic;
there exists a complex isotropic convex body K such that K is not (real) isotropic;
if K is complex isotropic and -invariant for every , then K is (real) isotropic.
Proof (i) From (2.5) and the definition of isotropy (2.3), we have
From (2.4), we also have .
(ii) We take K as
with , , . Then A is a right triangle and the origin is the center of mass of A. Therefore, K is convex body and the origin is the center of mass of K. Moreover, it follows that ,
In this example, it is easy to verify that (2.7) and (2.8) are true.
In order to verify (2.9), we divide it into two cases. For the case that ,
For the case that , together with (2.11), (2.12), we obtain
Therefore, K is complex isotropic. However, K is not (real) isotropic in view of (2.11) and (2.12).
(iii) From the assumption that K is a complex isotropic -invariant convex body in , we have . Note that and , together with (2.4), we obtain
From (2.3), it follows that . □
By Theorem 2.1, the class of complex isotropic bodies is larger than ones of real isotropic bodies.
3 The complex slicing problem
Observe that , together with Brunn’s theorem (see, e.g., [, p.18]), we have the following lemma.
Lemma 3.1 Let K be an origin-symmetric convex body in , and for . Then is concave and is decreasing for . Moreover, .
Lemma 3.2 Let K be a -invariant body in , then is constant for all .
Proof Since and for , we obtain
Obviously, there exists a map such that
for any satisfies .
Together with (3.1), we obtain
for any satisfies . □
Lemma 3.3 Let K be a -invariant complex isotropic body in with , then and , where .
Proof From (2.4) and Lemma 3.2, we have
A similar argument shows . □
The following lemma is given by Milman and Pajor in [, Lemma 2.1].
Lemma 3.4 Let be a measurable function such that and let L be a symmetric convex body in . Then the function
is an increasing function of p on .
Theorem 3.5 Let K be a -invariant complex isotropic convex body in with , then
Proof Let . From Lemma 3.2, it follows that is an even function. Note that K is -invariant implies K is origin symmetric. From Lemma 3.1, we have . Taking and in Lemma 3.4, combining with Lemma 3.3, we get
From the comparison , we have
we have equality in (3.2).
Lemma 3.6 Let be a decreasing function and let satisfying and such that Φ and are increasing. Then
is a decreasing function of p on (provided the integrals in are well defined).
Theorem 3.7 Let K be a -invariant complex isotropic convex body in with , then
Proof Let , . Let for , for and set . Note that K is -invariant implies K is origin symmetric. From Lemma 3.1, is convex and all hypotheses of Lemma 3.6 are satisfied, so by Lemma 3.3 we get
From the comparison , we have
is increasing when the positive integer n is increasing. Thus, the maximum of (3.4) occurs as n tends to infinity. □
4 Extremum of the complex isotropic constant of
The following lemma is proved in the spirit of the real counterpart (see, e.g., [, p.32]).
Lemma 4.1 Let , then
Proof Obviously, . We only need to consider the case that . Note that we identify with the real 2n-dimensional space equipped with the norm
On the other hand, we compute the same integral in polar coordinates and the polar formula for the volume:
Comparing these two expressions for the same integral, we get the result. □
Theorem 4.2 Let , then is a complex isotropic convex body in . Furthermore, its complex isotropic constant is
Proof Assume that . It is easy to verify that (2.7) and (2.8) are true for . Now, we only need to verify (2.9) for . Actually, we can explicitly calculate by using Lemma 4.1, to find that
Thus, () is complex isotropic.
Similarly, is complex isotropic and . □
Remark From Theorem 2.1(iii), is in fact isotropic since it is -invariant. However, the complex isotropic constant of is more useful as the argument in Section 3.
Next, we determine the extreme value of for . The approach that we adopted is similar to the real case due to Ma and the third named author .
Lemma 4.3 Let . Then, for each given positive integer n,
is a decreasing function for and an increasing function for .
Proof Making the change of variables , we have
It follows that
where . Now,
where we use the following integral representation for the function ψ:
and a change of variable .
Let in (4.2), then
Then it follows that
and, if ,
where we use the arithmetic-geometric mean inequality, i.e., .
Similarly, if , we have
The result follows from (4.1). □
The following lemma can be simply deduced as in [, p.4].
Lemma 4.4 For every complex isotropic convex body K in ,
for . Thus, from Lemma 4.4, we have
From Lemma 4.3 and (4.3), it follows that
for . The following lemma is given by Gao .
Lemma 4.5 Let and . For fixed x, the function
is a decreasing function of y for when and for when .
If we set and in (4.6) for , , by (4.3), it follows that
Combining with (4.5), we have
Together with (4.4) and (4.7), we have the following theorem.
Theorem 4.6 Let , then
Now we complete the proof of our main result.
Proof of Theorem 1.2 Combining with Theorem 3.5, Theorem 3.7, and Theorem 4.6, we obtain
The equality of the left inequality in this theorem holds for from the remark after Theorem 3.5 and (4.7). □
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The authors thank the referee for useful comments that helped to improve the presentation of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 11371239), Shanghai Leading Academic Discipline Project (Project Number: J50101), and the Research Fund for the Doctoral Programs of Higher Education of China (20123108110001).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
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Lai, D.N., Huang, Q. & He, B. Estimates for the extremal sections of complex -balls. J Inequal Appl 2014, 235 (2014). https://doi.org/10.1186/1029-242X-2014-235
- complex isotropic constant
- complex slicing problem