Open Access
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

© Wuhan University 2022

Licence Creative CommonsThis 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 . The layering function vanishes on functions whose support with empty interior and is strictly positive on functions whose support with non-empty interior. In this paper, we mainly prove the following theorems.

Theorem 1   Suppose that is a log-concave 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 log-concave 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 non-empty interior, then , with equality if only and if K and SuK 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 n-dimensional Euclidean space by R n and let Kn denote the set of all compact convex sets in R n . Let be the domain of log-concave function f, and let sub f denote the subgraph of f. Let denote the subgraph of f. For , let denote the n-dimensional 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 non-negative function is called a log-concave when the logarithm of the function is concave. Denote log-concave functionwhere is convex, and . It is clear that

Denote the level sets of log-concave 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 log-concave 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 origin-centered ellipsoids E1 and E2 in R n , if for any , the midpoints of the chords of E1 and E2 parallel to u are coplanar, then there exists r>0 such that

Lemma 3   (see Ref. [11]) If is a log-concave 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 non-empty interior, and that . Let denote the reflection of across the subspace . Since and has non-empty 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 .

Hence,

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 log-concave function , . This proves the result of Theorem 2.

References

  1. Steiner J . Einfache beweise der isoperimetrischen hauptsätze [J]. Journal für die reine und angewandte Mathematik (Crelles Journal), 1838, 1838(18): 281-296. [Google Scholar]
  2. Gardner J R . The Brunn-Minkowski inequality [J]. Bulletin (New Series) of the American Mathematical Society, 2002, 39 (3): 355-406. [Google Scholar]
  3. Bianchi G, Gronchi P. Steiner symmetrals and their distance from a ball [J]. Israel Journal of Mathematics, 2003, 135(1): 181-192. [CrossRef] [MathSciNet] [Google Scholar]
  4. Bianchi G, Klain D A, Lutwak E, et al. A countable set of directions is sufficient for Steiner symmetrization [J]. Advances in Applied Mathematics, 2011, 47(4): 869-873. [CrossRef] [MathSciNet] [Google Scholar]
  5. Haberl C, Schuster F E. General Lp affine isoperimetric inequalities [J]. Journal of Differential Geometry, 2009, 83(1): 1-26. [CrossRef] [MathSciNet] [Google Scholar]
  6. Klain D A . Steiner symmetrization using a finite set of directions [J]. Advances in Applied Mathematics, 2012, 48(2): 340-353. [CrossRef] [MathSciNet] [Google Scholar]
  7. Gruber P M . Convex and Discrete Geometry [M]. Berlin: Springer-Verlag, 2007. [Google Scholar]
  8. Webster R . Convexity [M]. Oxford: Oxford University Press, 1994. [MathSciNet] [Google Scholar]
  9. Bonnesen T, Fenchel W, Boron L F , et al. Theory of Convex Bodies [M]. Berlin: BCS Associates, 1987. [MathSciNet] [Google Scholar]
  10. Schneider R . Convex Bodies: The Brunn-Minkowski Theory [M]. Cambridge: Cambridge University Press, 2014. [Google Scholar]
  11. Lin Y J . Affine orlicz pólya-szegö principle for log-concave functions [J]. Journal of Functional Analysis, 2017, 273(10): 3295-3326. [CrossRef] [MathSciNet] [Google Scholar]

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