Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 2, April 2022



Page(s)  125  127  
DOI  https://doi.org/10.1051/wujns/2022272125  
Published online  20 May 2022 
Mathematics
CLC number: O186
Security Steiner’s Inequality on Layering Functions
School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing
400067, China
^{†} To whom correspondence should be addressed. Email: yjl@ctbu.edu.cn
Received:
15
December
2021
The classical definition of Steiner symmetrizations of functions are defined according to the Steiner symmetrizations of function level sets and the layered representation of functions. In this paper, the definition is not only transformed into Steiner symmetrizations of onedimensional parabolic functions, but also depends on the Steiner symmetrizations of the level sets of logconcave functions. To this end we prove Steiner’s inequality on layering functions in the space of logconcave functions.
Key words: convex body / Steiner symmetrizations / logconcave function
Biography: WEI Xinnan, female, Master candidate, research direction: convex geometric analysis. Email: 2020610034@email.ctbu.edu.cn
Foundation item: Supported by the National Natural Science Foundation of China (11971080) , Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202000838) , the Basic and Advanced Research Project of Chongqing (cstc2018jcyjAX0790, cstc2020jcyjmsxmX0328) and the Innovative Project of Chongqing Technology and Business University (yjscxx202111256)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Steiner symmetrizations were invented by Steiner^{[1]} to prove the isoperimetric inequality. For well over a century Steiner symmetrizations has played a fundamental role in answering questions about isoperimetry and related geometric inequalities^{[2]}. Steiner symmetrizations appears explicitly in the titles of numerous papers (see Refs. [3,4]) and plays a key roles in recent work such as Ref. [5].
Let be an open bounded subset of R ^{ n } For , we define the layering function :where r≥0 denotes by rB the closed Euclidean ball of radius r centered at the origin.
Evidently the function is monotonic and continuous on functions defined on Ω. The layering function vanishes on functions whose support with empty interior and is strictly positive on functions whose support with nonempty interior. In this paper, we mainly prove the following theorems.
Theorem 1 Suppose that is a logconcave function, and let u be a unit vector. Then its Steiner symmetrizations satisfies(1)The equality holds in (1) if and only if .
Theorem 2 Suppose that is a logconcave function. Then its symmetric decreasing rearrangement satisfies(2)The equality holds in (2) if and only if .
In Ref. [6], Klain proved geometric inequality which is corresponding with Theorem 1. If S(K) denotes the surface area of a compact convex set K having nonempty interior, then , with equality if only and if K and S_{u}K are translations. Theorem 1 is an extension of the geometric inequality by Ref. [6]. The layering function is more appropriate for our purposes, because the equality case in Theorem 1 is more stringent.
1 Preliminaries
Denote ndimensional Euclidean space by R ^{ n } and let K_{n} denote the set of all compact convex sets in R ^{ n }. Let Ω be the domain of logconcave function f, and let sub f denote the subgraph of f. Let denote the subgraph of f. For , let denote the ndimensional volume of K, and let u be a unit vector. Viewing K as a family of line segments parallel to u , slide these segments along u so that each is symmetrically balanced around the hyperplane , where denotes the complementary space of u . By Cavalieri’s principle, the volume of K is unchanged by this rearrangement. The new set, called the Steiner symmetrizations of K in the direction of u , will be denoted by . It is not difficult to show that is also convex, and that whenever . A little more work verifies the following intuitive assertion: if you iterate Steiner symmetrizations of K through a suitable sequence of unit directions, the successive Steiner symmetrals of K will approach a Euclidean ball in the Hausdorff topology on compact (convex) subsets of R ^{ n }. A detailed proof of this assertion can be found in Refs. [7, 8], for example.
In this section we present some specific elementary properties of Steiner symmetrizations, together with known facts to be used in the proof. Standard references for fundamental properties of Steiner symmetrizations are the Ref. [9]. See also Ref. [10] for general background on classical convexity theory.
A nonnegative function is called a logconcave when the logarithm of the function is concave. Denote logconcave functionwhere is convex, and . It is clear that
Denote the level sets of logconcave function:
Then is convex. By the layer representation,(3)
For , we define the symmetric decreasing rearrangement of f by symmetrizing its level sets, that is
Given a compact convex set K and a unit vector u , we have (or respectively, up to translation) if and only if K is symmetric under reflection across the subspace (respectively, up to translation). In particular, will hold for every direction u (or even a dense set of directions) if and only if K is a Euclidean ball centered at the origin.
Given compact convex subsets and , we denote
An expression of this form is called a Minkowski combination or Minkowki sum. Since K and L are convex sets, the set aK+bL is also convex. Convexity also implies that aK+bK=(a+b)K for all a,b≥0, although this does not hold for general sets.
2 Proof of the Main Theorem
The following crucial property of Steiner symmetrizations will be used in the main theorems.
Lemma 1 (see Ref. [11]) If is a logconcave function, then there exists a sequence of successive Steiner symmetrizations of f which approximates its symmetric in the norm , i.e.,
Lemma 2 (see Ref. [11]) For two origincentered ellipsoids E_{1} and E_{2} in R ^{ n }, if for any , the midpoints of the chords of E_{1} and E_{2} parallel to u are coplanar, then there exists r>0 such that
Lemma 3 (see Ref. [11]) If is a logconcave function, andThen .
Lemma 4 If D is a ball centered at the origin, and if X is a line segment, parallel to the unit vector u , having one endpoint in the interior of D and the other endpoint outside D, then Steiner symmetrizations will strictly increase the slice length; that is(4)
Proof Let denote the line through X. Our conditions on the endpoints of X imply that . Meanwhile, fixes D and slides X parallel to u until it is symmetric about . If , then will lie wholly inside D, so that and (4) follows. If , then will cover the slice completely, so that and (4) follows once again.
Proof of Theorem 1 Let u be a unit vector. The monotonicity of Steiner symmetrizations implies thatThuswhence . It follows that
Evidently equality holds if . For the converse, suppose that has nonempty interior, and that . Let denote the reflection of across the subspace . Since and has nonempty interior, there is a point such that . Let D denote the ball around the origin of radius , and let denote the line through x and parallel to u . The slice meets the boundary of D at x on one side of , and has an endpoint outside D and another endpoint in the interior of D, where . It follows from (4) of Lemma 4 that
Moreover, this holds for parallel slices through point x' in an open neighborhood of x. After integration of parallel slice lengths to compute volumes, we obtainfor values of r in an open neighborhood of x. It follows that .
So, if , then for any h>0. Therefore, by (3) that .
Therefore, we complete the proof of Theorem 1.
Proof of Theorem 2 Suppose . By Lemma 2 and Lemma 3, there exist some direction u such that the midpoints of the chords of sub f parallel to u do not lie in any linear subspace of parallel to . Let . From Theorem 1 it follows that
By Lemma 1, there exists a sequence of directions , such that the sequence defined by converges to in the norm. Thus, by the continuity of in the space , we have
This implies that if , then . Using the same argument, we can get that for any logconcave function , . This proves the result of Theorem 2.
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