Estimates for the extremal sections of complex -balls
© Lai et al.; licensee Springer. 2014
Received: 28 November 2013
Accepted: 29 May 2014
Published: 6 June 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.
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.
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 .
as for each .
The orthogonal two-dimensional subspace has an orthonormal basis ξ, .
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]).
for every .
for all .
Now, we show the relation between real isotropic bodies and complex isotropic bodies when we consider the convex body defined in .
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.
From (2.4), we also have .
In this example, it is easy to verify that (2.7) and (2.8) are true.
Therefore, K is complex isotropic. However, K is not (real) isotropic in view of (2.11) and (2.12).
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 .
for any satisfies .
for any satisfies . □
Lemma 3.3 Let K be a -invariant complex isotropic body in with , then and , where .
A similar argument shows . □
The following lemma is given by Milman and Pajor in [, Lemma 2.1].
is an increasing function of p on .
we have equality in (3.2).
is a decreasing function of p on (provided the integrals in are well defined).
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]).
Comparing these two expressions for the same integral, we get the result. □
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 .
is a decreasing function for and an increasing function for .
and a change of variable .
where we use the arithmetic-geometric mean inequality, i.e., .
The result follows from (4.1). □
The following lemma can be simply deduced as in [, p.4].
for . The following lemma is given by Gao .
is a decreasing function of y for when and for when .
Together with (4.4) and (4.7), we have the following theorem.
Now we complete the proof of our main result.
The equality of the left inequality in this theorem holds for from the remark after Theorem 3.5 and (4.7). □
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).
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