﻿ 凸体的Steiner对称化的两个定理 Two Theorems of Steiner Symmetrization on Convex Bodies

Pure Mathematics
Vol.07 No.05(2017), Article ID:21748,5 pages
10.12677/PM.2017.75047

Two Theorems of Steiner Symmetrization on Convex Bodies

Liying Sun

School of Mathematics and Information Science, Shaanxi Normal University, Xi’an Shaanxi

Received: Aug. 1st, 2017; accepted: Aug. 16th, 2017; published: Aug. 21st, 2017

ABSTRACT

In this paper, we mainly study sufficient conditions for Steiner symmetrization on convex bodies. Firstly, according to the properties of Steiner symmetrization, such as volume-preserving, convexity-preserving, monotonicity, surface area reduction and so on, we constructed a transformation on convex bodies. Secondly, in accordance to Steiner symmetrization’s characterization and concept, we proved that is Steiner symmetrization and came up with two homologous corollaries. Finally, we obtained two sufficient conditions for Steiner symmetrization.

Keywords:Convex Body, Steiner Symmetrization, Minkowski Symmetrization, Hyperplane

1. 引言

1836年，Jakob Steiner介绍了凸体Steiner对称化的过程，并利用Steiner对称化尝试着证明了经典的等周不等式和经典的Brunn-Minkowski不等式，这在 [1] 中第九章有详细证明过程。凸体在做Steiner对称化后，它还有些重要的性质，例如：体积不变、保凸性、单调性、直径不增等，这些性质在 [2] [3] [4] [5] [6] 中可以查看，以及一般的紧集也可以做Steiner对称化，这可以在 [7] 中查看。除了在几何方向有着重要作用外，Steiner对称化还在分析及PDE方向起着重要的作用。

2. 预备知识

Minkowski对称有些好的性质，它与Steiner对称化有一些好的关系 [5] [10] 。

3. 定理的证明

1)

2)

3)

4)关于对称，

，设，则：

.

.

1)

2)

3)

4)

1)

2)

3)

4)关于对称，

1)

2)，有

3)

4)

，这里都为紧(闭)凸集。又由(1)知，，故有

4. 结语

Two Theorems of Steiner Symmetrization on Convex Bodies[J]. 理论数学, 2017, 07(05): 368-372. http://dx.doi.org/10.12677/PM.2017.75047

1. 1. Gruber, P. M. (2007) Convex and Discrete Geometry. Springer, Berlin, 168-179, 120-132.

2. 2. 刘越. Steiner对称化及平面上经Steiner对称化保持直径的凸集[D]: [硕士学位论文]. 广州: 中山大学, 2009.

3. 3. Klain, D.A. (2011) Steiner Symmetrization Using a Finite Set of Directions. Advances in Applied Mathematics, 48, 340-353. https://doi.org/10.1016/j.aam.2011.09.004

4. 4. Coupier, D. and Davydov, Y. (2014) Random Symmetrizations of Convex Bodies. Advances in Applied Probability, 46, 603-621. https://doi.org/10.1017/S000186780000728X

5. 5. Bourgain, J., Lindenstrauss, J. and Milman, V. (1989) Estimates Related to Steiner Symmetrizations. In: Lindenstrauss, J. and Milman, V.D., Eds., Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, Vol. 1376, Springer, Berlin, Heidelberg, 264-273. https://doi.org/10.1007/BFb0090060

6. 6. Schneider, R. (1993) Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge. https://doi.org/10.1007/BFb0090060

7. 7. Falconer, K.J. (1976) A Result on the Steiner Symmetrization of a Compact Set. Journal of the London Mathematical Society, S2-14, 385-386. https://doi.org/10.1112/jlms/s2-14.3.385

8. 8. Bianchi, G., Gardner, R.J. and Gronchi, P. (2017) Symmetrization in Geometry. Advances in Mathematics, 306, 51-88. https://doi.org/10.1016/j.aim.2016.10.003

9. 9. Krantz, S.G. and Parks, H.R. (1999) The Geometry of Domains in Space. Birkhauser, Basel, 223-246. https://doi.org/10.1007/978-1-4612-1574-5

10. 10. Klartag, B. (2000) Remarks on Minkowski Symmetrizations. In: Milman, V.D. and Schechtman, G., Eds., Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, Vol. 1745, Springer, Berlin, Heidelberg, 109-117. https://doi.org/10.1007/BFb0107211