凸体几何,几何分析。

2003. 07 --- 2015. 03 Department of Mathematics, Shanghai University ;

Associate Professor (2007/03), Professor (2012/03), Ph. D. Supervisor (2013/05).

Since 2015. 04 Professor, Department of mathematics, Tongji University;

2007--- 2018 Visiting Research Associate, The University of Hongkong;

2008. 09 --- 2009. 09 Visiting scholar, New York Universty, USA;

2013/06/20---2013/07/10 Visiting scholar, Chern Institute of Mathematics, Nankai University;

2014/07/01---2014/07/16 Visiting scholar, Chern Institute of Mathematics, Nankai University;

2017/07/23---2017/08/05 Visiting scholar, Chern Institute of Mathematics, Nankai University.

Selected Publications

27.Sharp affine isoperimetric inequalities for the volume decomposition functionals of polytopes，Advances in Mathematics,389 (2021), 107902.

26.On the continuity of the solutions to the Lp capacitary Minkowski problem,Proceedings of the AMS,149 (2021), 3063–3076.

25. Steiner symmetrization (n -1) times is sufficient to transform an ellipsoid to a ball in R^n,Annales mathématiques du Québec,45 (2021), 221–228.

24.The Lp Minkowski problem for the electrostatic capacity,Journal of Differential Geometry,116 (2020), 555-596.

23.The Lp Minkowski problem for the electrostatic /mathfrak{p}-capacity for/mathfrak{p} > n,Indiana Univ. Math. J., 70 (2021), 1869-1901.

22.The Orlicz Brunn-Minkowski inequality for the projection body,Journal of Geometric Analysis, 30 (2020),2253-2272.

21. Sharp affine isoperimetric inequalities for the Minkowski first mixed volume, Bulletin of the London Mathematical Society,52 (2020),161-174.

20. Affine isoperimetric inequalities for intersection mean ellipsoids,Calculus of Variations and PDEs, (2019), 58: 191.

19. A new approach to the Minkowski first mixed volume and the LYZ conjecture,Discrete Comput. Geom.,66(2021), 122–139.

18. The Lp capacitary Minkowski problem for polytopes，Journal of Functional Analysis, 277 (2019), 3131-3155.

17. A new affine invariant geometric functional for polytopes and its associated affine isoperimetric inequalities,International Mathematics Research Notices, (2021), no. 12, 8977–8995.

16. New affine inequalities and projection mean ellipsoids,Calculus of Variations and PDEs, (2019), 58: 44, 18 pp.

15. Extremal problems for Lp surface areas and John ellipsoids,Journal of Mathematical Analysis and Applications, 479 (2019), 1226-1243.

14. On mixed Lp John ellipsoids,Advances in Geometry, 19 (2019),297-312.

13. The logarithmic John ellipsoid,Geometriae Dedicata, 197 (2018), 33–48.

12. Convex bodies with identical John and LYZ ellipsoids,International Mathematics Research Notices, 2018 (2018), 470-491.

11. A unified treatment for Lp Brunn-Minkowski type inequalities, Communications in Analysis and Geometry, 26 (2018), 435-460.

10. Orlicz-Legendre ellipsoids,Journal of Geometric Analysis, 26 (2016) , 2474-2502.

9. Orlicz-John ellipsoids,Advances in Mathematics, 265 (2014), 132-168.

8. Extremum problems for the cone volume functional of convex polytopes, Advances in Mathematics, 225 (2010), 3214-3228.

7. The minimal Orlicz surface area, Advances in Applied Mathematics, 61 (2014), 25-45.

6. Bounds for inclusion measures of convex bodies,Advances in Applied Mathematics, 41 (2008), 584--598.

5. Orlicz mixed quermassintegrals, Science China Mathematics, 57 (2014), 2549-2562.

4. Chord power integrals and radial mean bodies,J. Math. Anal. Appl., 342 (2008), 629-637.

3. Orlicz mixed affine quermassintegrals,Science China Mathematics, 58 (2015), 1715-1722.

2. Inequalities for chord power integrals,J. Korean Math. Soc., 45 (2008), 587-596.

1. Reconstructing triangles inscribed in convex bodies from X-ray functions,Acta Math. Sci. Ser. B, 24 (2004), 608-612.

LectureNotes and Surveys

1. 凸体几何与泛函分析，83 pages, 胡家麒，熊革。

2. 凸体几何中的 Minkowski 问题，70 pages, 鲁新宝，熊加威，熊革。

1.Extremum problems for the cone volume functional of convex polytopes, Advances in Mathematics, 225 (2010), 3214-3228.

Abstract: Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in R^n containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R^2 and R^3. Some new sharp inequalities characterizing parallelotopes in R^n are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in R^n is provided.

简介：2001 年，Lutwak, D. Yang, G. Zhang 在Transactions AMS 上发表的文章（A new affine invariant for polytopes and schneider's projection problem，353（2001），1767-1779）对 n 维欧氏空间中的多胞形引入了一个新的仿射不变量， 并且提出了关于这个仿射不变量与多胞形体积比值的极值问题猜想。本文 解决了Lutwak - Yang - Zhang 猜想当空间维数 n= 2,3 的情形。

它引情况（根据 MathSciNet）：被发表在期刊 JAMS（1 篇），JDG（3 篇）， Adv. Math.（7 篇）, TAMS （1 篇）, IMRN （1 篇）, PAMS, Monatsh. Math., Adv. Appl. Math. (3篇) 等上面的文章多次它引和正面评价。

2. Orlicz-John ellipsoids,Advances in Mathematics, 265 (2014), 132-168.

Abstract: The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved Lp John ellipsoids. The analog of Ball's volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.

简介：本文把 凸体几何 中的重要几何体 --- John 椭球, 推广到了Orlicz Brunn-Minkowski 理论框架下。

这个工作是在 Orlicz Brunn-Minkowski 理论自 2010 年研究兴起的背景下，我们 解决的 Orlicz BM 理论中的一个基本问题。经典的 John 椭球由 F. John 于 1948 年引入，在 PDE, 泛函分析，优化问题中有诸多应用。在凸体几何中，John 定理与泛函分析中著名的 Brascamp-Lieb 不等式结合起来是建立反向的仿射等周不等式的有效工具。

2005 年， E. Lutwak，D. Yang 和 G. Zhang 等在 Proceedings LMS 上发表的文章（Lp John ellipsoids, 90 (2005), 497-520）把凸体几何中两个基本重要性的几何体 --- John 椭球和 LYZ 椭球统一到了 Lp Brunn-Minkowski 理论框架下，定义了 Lp John 椭球的概念并建立了 Lp 版本的 Keith Ball 体积比不等式。如何把它们进一步推广到 Orlicz BM 理论框架下， 是凸体几何自 2010 年以来的一个基本问题, 而问题的困难之处在于 Orlicz 混合体积缺少齐次性。

我们通过引入“拟 Orlicz 混合体积”这一概念，证明由 Orlicz 混合体积与 拟 Orlicz 混合体积定义的椭球是一致的（除去常数倍）, 得到了John 椭球的满意推广，并且建立了Orlicz 情形下几个新的仿射等周不等式， 包括 Orlicz 版本的 Keith Ball 体积比不等式。 在文章中，我们还刻划了凸体的 Orlicz John 椭球是球与其 Orlicz 表面积测度迷向性之间的联系。

它引情况（根据 MathSciNet）：被发表在期刊 JDG（1 篇）, Adv. Math.（6 篇）, CVPDE（2 篇）, JFA（1 篇）, IUMJ（1 篇）, PAMS，IMRN, Monatsh. Math., Adv. Appl. Math.（2 篇）, JGEA 等上面的文章多次它引。

3. A unified treatment for Lp Brunn-Minkowski type inequalities, Communications in Analysis and Geometry, 26 (2018), 435-460.

Abstract: A unified approach used to generalize classical Brunn-Minkowski type inequalities to Lp Brunn-Minkowski type inequalities, called theLp transference principle, is refined in this paper. As illustrations of the effectiveness and practicability of this method, several new Lp Brunn-Minkowski type inequalities concerning the mixed volume, moment of inertia, quermassintegral, projection body and capacity are established.

简介：本文提出并证明了一个新原理：若定义在凸体类上的非负泛函是正齐次单调增且凹的，则该泛函是 p-凹的。 进一步的，若泛函还是严格增的，则不等式中等号成立当且仅当两个凸体互为膨胀。作为该原理有效性的应用和实证，本文建立了几个新的 Lp Brunn-Minkowski 型不等式。这其中包括把著名数学家 C. Borell (Math. Ann. 263 (1983), 179-184.), L. Caffarelli, D. Jerison 和 E. Lieb (Adv. Math. 117 (1996), 193-207.) 建立的电容的 Brunn-Minkowski 型不等式推广到了 Lp 情形。

此结果受到审稿专家的高度评价。譬如，“... Of the new inequalities obtained I might single out the Lp version of the general capacitary Brunn-Minkowski inequalityas a particular nice consequence of this new principle. … I find itremarkablethat the main result, Theorem 1.1, has not been published before (as far as I am aware) . … I am sure it will be used and cited very often in the future. ”

4. Convex bodies with identical John and LYZ ellipsoids,International Mathematics Research Notices, 2018 (2018), 470-491.

Abstract:Convex bodies with identical John and LYZ ellipsoids are characterized. This solves an important problem from convex geometry posed by G. Zhang. As applications, several sharp affine isoperimetric inequalities are established.

简介：本文解决了凸体几何中的一个重要问题：当凸体满足什么条件时， 其John 椭球与 LYZ 椭球是一致的？ 这个问题由著名凸体几何学家、纽约大学张高勇（Zhang Gaoyong）教授 2000 年提出。

John 椭球是凸体几何中的一个基本几何体，尤其是 John 特征定理与分析中著名的 Brascamp-Lieb 不等式结合起来在建立反向的仿射等周不等式中起到了关键的作用。LYZ 椭球由纽约大学 E. Lutwak, D. Yang 和 G. Zhang 三位教授2000年合作发表在 Duke Math. J. 上的文章首先发现而得名，有趣的是它与经典力学中的惯性椭球（又叫Legendre 椭球）是对偶的，而且与 Fisher 信息矩阵密切相关。

本文彻底解决了上述问题，给出了几个简洁的充要条件，并且证明了所分离出的凸体类在 Hausdorff 度量下是完备的。作为应用，我们推广了 F.Schuster 与 M.Weberndorfer 2012年合作发表在 J. Differential Geometry 上的工作， 建立了3个严格的反向仿射等周不等式。

这一工作受到审稿专家的高度好评。譬如，“... The result gives a simple necessary and sufficient condition for convex bodies with identical John and LYZ ellipsoids.It is beautiful and important…”；“... The paper under review iswithout a doubt a very important contributionto convex geometric analysis. Its two main results willundoubtedly be tremendously usefulto future researchers working on problems from the Lp theory as well as the Orlicz theory of convex bodies. …”。

5. New affine inequalities and projection mean ellipsoids, Calculus of Variations and PDEs, 58 (2019), Art. 44, 18 pp.

Abstract: A variational formula for the Lutwak affine surface areas A j of convex bodies in R^n is

established when 1 ≤ j ≤ n − 1. By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for A j , which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.

简介：在这篇文章中，我们首先建立了积分仿射表面积 (Integral affine surface area) 的变分公式，定义了变分公式中所产生的新测度：仿射投影测度 (Affine projection measure)。然后我们创造性地结合了凸体几何中重要的几何手段--- 投影函数 （Projection function）与 研究逆仿射等周问题的重要工具---John 椭球 (John ellipsoid) 这两个基本要素，定义并证明了投影平均椭球 (Projection mean ellipsoid) 的存在唯一性，由此建立了积分仿射表面积的严格的等周型不等式。

这提供了研究凸体几何中 30 余年未取得任何进展的 Lutwak 仿射表面积猜想 ( E. Lutwak, Inequalities for Hadwiger's harmonic quermassintegrals. Math. Ann. 280, 165–175 (1988)) 的全新思路。

6. Sharp affine isoperimetric inequalities for the volume decomposition functionals of polytopes,Advances in Mathematics,389 (2021), 107902.

Abstract:The nth power of thevolume functionalof polytopes P in R^n, according todimensionsof the spaces spanned by any n unit outer normal vectors of P, is decomposed inton homogeneous polynomialsof degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R^3 are established, which essentially characterize the geometric and algebraic structures of polytopes.

简介：在这篇文章中，我们有一个基本的发现：利用线性代数中线性无关(linear independence)的概念以及数学中维数(dimension)的概念，把多面体的体积(volume)这个基本的几何量分解成n个n次齐次多项式。其中第n个多项式恰好是2001年纽约大学的E. Lutwak, D. Yang和G. Zhang教授引入的仿射不变量（TAMS，353 (2001), 1767-1779）.

本文完全解决了R^3中体积分解泛函的极值问题，建立了严格的仿射反向等周不等式，刻画了平行六面体。

1. Brief Introduction on Convex Geometry:

In mathematics,convex geometryis a branch of geometry that studies geometric structures and invariants of convex sets by using both geometric and analytic methods, mainly in Euclidean space. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three.

The Brunn-Minkowski theory, also known as the theory of mixed volumes lies at the very core of convex geometry. Main results in the Brunn-Minkowski theory include the Brunn-Minkowski inequality, the Aleksandrov-Fenchel inequality, the solution of the Minkowski problem, and Hadwiger's characterization theorem.

Besides the theory of mixed volumes, affine geometry of convex sets is another integral part of convex geometry. Affine isoperimetric inequalities and reverse affine isoperimetric inequalities play a dominant role in affine geometry of convex sets.

During the last two decades, convex geometry has achieved important developments. Lutwak's dual Brunn-Minkowski theory has obtained major breakthroughs by studying the celebrated Busemann-Petty problem. As extensions of the classic Brunn-Minkowski theory, the Lp Brunn-Minkowski theory and then the Orlicz Brunn-Minkowski theory are developing rapidly.

2. Research Grants:

(1) NSFC for the Youth (Grant No. 11001163);

(2) NSFC for the General Program (Grant No. 11471206)；

(3） NSFC for the General Program (Grant No. 11871373).

一. 本科教学:

• 2015 --- 2016 学年 第一学期 数学分析(上) 12210902； 数学分析（上）习题课 12222801.

• 2015 --- 2016 学年 第二学期 数学分析(中) 12211002； 数学分析（中）习题课 12222901.

• 2016 --- 2017 学年 第一学期 数学分析(下) 12211101； 数学分析（下）习题课 12225402.

• 2016 --- 2017 学年 第二学期 高等数学(B)下12200525.

• 2018 --- 2019 学年 第一学期 实变函数 12212201.

• 2018 --- 2019 学年 第二学期 复变函数 12222102.

• 2019 --- 2020 学年 第一学期 数学分析(上) 12210902； 数学分析（上）习题课 12222802.

• 2019 --- 2020 学年 第二学期 数学分析(中) 12211001； 数学分析（中）习题课 12222902.

• 2020 --- 2021 学年 第一学期 数学分析(下) 12227501； 数学分析（下）习题课 12225401.

• 2020 --- 2021学年第二学期复变函数12222101.

• 2021 --- 2022 学年 第一学期 常微分方程 12222002.

二. 指导博士生：

1. 已经毕业

（1） 邹都， 2015 年 06 月。

博士论文题目：Orlicz Brunn-Minkowski 理论中的仿射极值问题。

工作单位：武汉科技大学数学系。

（2） 胡家麒 , 2018 年 06 月。

博士论文题目：几何测度与仿射等周型问题。

工作单位：杭州师范大学数学系。

（3）熊加威, 2021年06月。

博士论文题目：凸几何中静电容量的Minkowsk问题研究。

工作单位：宁波大学数学与统计学院。

2. 在读

（1） 鲁新宝

（2） 刘裕德

（3） 孙  强

（4） 陶江艳

（5） 陈  章

三. 博士后

(1) 汪卫

工作单位 : 湖南科技大学数学系。