席东盟

席东盟

Dongmeng Xi

上海大学理学院数学系教授，伟长学者。研究凸几何与积分几何中的分析问题：Mahler 猜想、Dar 猜想与 log-Brunn–Minkowski 猜想，几何测度的 Minkowski 问题，以及与之对应的 Monge–Ampère 型方程。 Professor of Mathematics at Shanghai University and Weichang Scholar. Works on analytic problems in convex and integral geometry: the Mahler conjecture, Dar's conjecture and the log-Brunn–Minkowski conjecture, Minkowski problems for geometric measures, and the associated Monge–Ampère type equations.

代表论文Selected papers 联系方式Get in touch
单形与它的极体Simplex and its polar
立方体与八面体Cube and octahedron
椭球与它的极体Ellipsoid and its polar
平面常宽体：Reuleaux 三角形Constant width in the plane: Reuleaux triangle
八面体的投影体：菱形十二面体Projection body of the octahedron: rhombic dodecahedron
三维常宽体：Meissner 体Constant width in space: Meissner body

研究方向Research

研究属于几何分析，主要围绕凸几何与积分几何中的分析问题。成果发表于 Comm. Pure Appl. Math.、J. Eur. Math. Soc.、J. Differential Geom.、Adv. Math.、Math. Ann.、Trans. Amer. Math. Soc.、J. Funct. Anal. 等期刊。 Geometric analysis, centred on analytic problems in convex and integral geometry. Work has appeared in Comm. Pure Appl. Math., J. Eur. Math. Soc., J. Differential Geom., Adv. Math., Math. Ann., Trans. Amer. Math. Soc., J. Funct. Anal. and elsewhere.

V(K)V(K°) —
64/9(4π/3)²
三维：K（蓝）在椭球与四面体之间连续变化，琥珀色为极体 K°。可以拖动旋转。3D: K (blue) moves between an ellipsoid and a tetrahedron; amber is the polar K°. Drag to rotate.
V(K)V(K°) —
32/3(4π/3)²
三维对称情形：K（蓝）在立方体与椭球之间连续变化，琥珀色为极体 K°。The symmetric case in 3D: K (blue) moves between a cube and an ellipsoid; amber is the polar K°.

Mahler 猜想The Mahler conjecture

Mahler 猜想起源于 1938 年：凸体 K 与极体 K° 的体积乘积，在原点对称情形下猜想由立方体取到最小值；一般情形下，K 内部含原点，猜想由质心在原点的单纯形取到最小值。Mahler 证明了这两个不等式的二维情形。A. D. Aleksandrov 于 1967 年独立提出了一般情形的猜想。 The Mahler conjecture dates back to 1938: the volume product of K and its polar K° is conjectured to be minimised by cubes in the origin-symmetric case, and by simplices with centroid at the origin in the general case, where K contains the origin in its interior. Mahler proved both inequalities in dimension two. A. D. Aleksandrov independently formulated the general case in 1967.

2026 年与陈世炳、李媛媛、徐哲锋合作，建立了 shadow flow 方法，解决了三维的一般情形。 A 2026 preprint with Shibing Chen, Yuanyuan Li and Zhe-Feng Xu develops the shadow-flow method and settles the general case in dimension three.

三维一般情形General case, dimension 3V(K) V(K°)≥649

同一框架还给出了三维对称情形的新的纯几何证明；该情形由 Iriyeh 与 Shibata 首先证明。 The same framework gives a new, purely geometric proof of the symmetric case in dimension three, first proved by Iriyeh and Shibata.

三维对称Symmetric, dimension 3V(K) V(K°)≥323
  • Chen, Li, Xi, Xu. The Mahler conjecture in three dimensions. arXiv:2605.09334 (2026)
多面体的 Minkowski 问题：每个面上的箭头是 aivi（外法向乘以面积）。动画中箭头离开多面体、首尾相接，恰好回到起点。可以拖动旋转。The Minkowski problem for polytopes: each face carries the arrow aivi (outer normal times area). In the animation the arrows leave the polytope and are laid head to tail; they close up exactly. Drag to rotate.
随机直线截出的弦Random chords through a convex body

积分几何与仿射几何中的 Minkowski 问题Minkowski problems in integral and affine geometry

Minkowski 定理（1897）：给定张成 ℝn 的互异单位向量 vi 与正数 ai，以它们为外法向和面面积的凸多面体存在，当且仅当 Minkowski's theorem (1897): for distinct unit vectors vi spanning ℝn and positive ai, a convex polytope with these outer normals and face areas exists if and only if

Minkowski 条件Minkowski's conditiona1v1+⋯+aNvN=0

把面积换成其他几何测度，就得到新的 Minkowski 问题。与 Lutwak、Yang、张高勇将 Aleksandrov 变分法引入积分几何，由弦积分（下式）导出弦测度；又与合作者研究了仿射面积测度、仿射对偶测度与中心截面测度。 Other measures in place of area give new Minkowski problems. With Lutwak, Yang and Zhang he brought Aleksandrov's variational method to integral geometry, deriving chord measures from the integrals below; later work treats affine and centro-section measures.

Iq(K)=∫𝓛n|K ∩ ℓ|q dℓ
  • Lutwak, Xi, Yang, Zhang. Chord measures and Minkowski problems. Comm. Pure Appl. Math. 77 (2024)
  • Xi, Zhao. Fractional affine area measures. J. Eur. Math. Soc., to appear
  • Cai, Leng, Wu, Xi. Affine dual Minkowski problems. Adv. Math. 467 (2025)
  • Cai, Leng, Wu, Xi. Minkowski problems of centro-section measures. Adv. Math. 486 (2026)
平面：深色为 K 与 x + L 的最大重叠 M(K, L)；下方比较 Dar 不等式两边与 Brunn–Minkowski 下界。The plane: the dark region is the largest overlap M(K, L); the bars compare both sides of Dar's inequality with the Brunn–Minkowski bound.
V(K+L)1/3 —
右边right side —
三维取等：K = C + I，L = aC + a−1I。拖动 a，两边始终相等。3D equality: K = C + I, L = aC + a−1I. Drag a: the two sides stay equal.
log 组合 (1−λ)·K +0 λ·L（K、L 处于 dilation position），实时验证 log-Brunn–Minkowski 不等式The log-combination (1−λ)·K +0 λ·L of K, L at a dilation position, with the log-Brunn–Minkowski inequality checked live

Dar 猜想与 log-Brunn–Minkowski 猜想Dar's conjecture and the log-Brunn–Minkowski conjecture

Dar（1999）猜想 Brunn–Minkowski 不等式可加强为下式，其中 M(K, L) = maxx V(K ∩ (x + L))。Xi–Leng 引入 dilation position，完整解决了平面情形。 Dar (1999) conjectured the strengthening of Brunn–Minkowski below, where M(K, L) = maxx V(K ∩ (x + L)). Xi–Leng introduced the dilation position and settled the planar case.

V(K + L)1n≥M(K, L)1n+V(K)1nV(L)1nM(K, L)1n

同一方法给出处于 dilation position 的平面凸体的 log-Brunn–Minkowski 不等式（下式），回答了 Böröczky–Lutwak–Yang–Zhang 的问题；2026 年的预印本给出两者统一的简短证明。 The same idea proves the log-Brunn–Minkowski inequality below for planar convex bodies in dilation position, answering a question of Böröczky, Lutwak, Yang and Zhang; a 2026 preprint unifies both proofs.

V((1−λ)·K +0 λ·L)≥V(K)1−λ V(L)λ
  • Xi, Leng. Dar’s conjecture and the log-Brunn–Minkowski inequality. J. Differential Geom. 103 (2016)
  • Xi. A very short unified proof of Dar’s conjecture and log-Brunn–Minkowski in the plane. arXiv:2609.08495 (2026)
  • Xi. The reverse-log-Brunn–Minkowski inequality. arXiv:2307.04266
绿色为 Orlicz 和 K +φ L（φ(t) = tp），它始终包含 K 与 L；左下角实时验证不等式Green: the Orlicz sum K +φ L with φ(t) = tp; it always contains K and L. Bottom left: the inequality, checked live

Orlicz Brunn–Minkowski 理论Orlicz Brunn–Minkowski theory

引入内部含原点的凸体的 Orlicz 加法 K +φ L（φ 为凸且严格递增函数，φ(0) = 0），并建立了 Orlicz Brunn–Minkowski 不等式（下式）；取 φ(t) = tp 即回到 Lp 情形。后续工作包括离散 Orlicz Minkowski 问题。 Introduced Orlicz addition K +φ L of convex bodies containing the origin in their interiors (φ convex and strictly increasing, φ(0) = 0) and proved the Orlicz Brunn–Minkowski inequality below; φ(t) = tp recovers the Lp case. Follow-up work treats the discrete Orlicz Minkowski problem.

φ(V(K)1nV(K +φ L)1n) + φ(V(L)1nV(K +φ L)1n) ≤ 1
  • Xi, Jin, Leng. The Orlicz Brunn–Minkowski inequality. Adv. Math. 260 (2014)
  • Wu, Xi, Leng. On the discrete Orlicz Minkowski problem. Trans. Amer. Math. Soc. 371 (2019)
两个六边形的高斯表面积测度完全相同（箭头）；只有 γ ≥ 1/2 的那个落在唯一性范围内Two hexagons with exactly the same Gaussian surface area measure (arrows); only the one with γ ≥ 1/2 lies in the uniqueness class

高斯概率空间中的 Minkowski 问题The Minkowski problem in Gaussian space

Huang–Xi–Zhao 提出了高斯表面积测度（下式）的 Minkowski 问题。高斯测度既不平移不变也不齐次，解可以不唯一（左图两个六边形的测度相同）；他们证明在 γ(K) ≥ 1/2 的凸体中解唯一，并得到适当的偶数据下的存在性。 Huang–Xi–Zhao posed the Minkowski problem for the Gaussian surface area measure (below). Gaussian measure is neither translation invariant nor homogeneous. Solutions need not be unique (the two hexagons have the same measure); they proved uniqueness when γ(K) ≥ 1/2 and existence for suitable even data.

Sγ(K, ω)=1(√2π)n∫νK−1(ω)e−|x|²2 dℋn−1
  • Huang, Xi, Zhao. The Minkowski problem in the Gaussian probability space. Adv. Math. 385 (2021)

学术交流Talks and visits

受邀报告与学术访问。Invited talks and research visits.

受邀报告Invited talks

  • 2027.01
    国际华人数学家大会（ICCM 2026）International Congress of Chinese Mathematicians (ICCM 2026)大会报告Plenary即将举行Upcoming
  • 2026.08
    Prague Conference on Convex Geometry大会报告Plenary
  • 2026.01
    国际华人数学家大会（ICCM 2025）International Congress of Chinese Mathematicians (ICCM 2025)大会报告Plenary
  • 2024.12
    Convex Geometry and its Applications德国 Oberwolfach 数学研究所Mathematisches Forschungsinstitut Oberwolfach, Germany
  • 2018.12
    Convex Geometry and its Applications德国 Oberwolfach 数学研究所Mathematisches Forschungsinstitut Oberwolfach, Germany

学术访问Research visits

  • 2014–2015
    纽约大学，联合培养博士New York University, joint PhD training师从 Lutwak、Yang 与张高勇With Erwin Lutwak, Deane Yang and Gaoyong Zhang
  • 2019–2021
    纽约大学 Courant 数学研究所，访问学者Courant Institute, New York University, visiting scholar
  • 2022–2026
    多次赴欧洲及美国学术访问Repeated research visits to Europe and the United States欧洲：维也纳工业大学、柏林工业大学、萨尔茨堡大学Europe: Technische Universität Wien; Technische Universität Berlin; Paris Lodron Universität Salzburg美国：纽约大学 Courant 数学研究所、雪城大学USA: Courant Institute, New York University; Syracuse University

报告视频Recorded talks

    视频托管于 YouTube；中国大陆访问可能需要相应网络环境。Videos are hosted on YouTube.

    学术服务Professional service

    组织学术会议，并为国际期刊审稿。Conference organisation and refereeing.

    组织的学术会议Conferences organised

    • 2026.06
      Brunn–Minkowski Theories: 庆祝 Erwin Lutwak 80 岁生日80th birthday of Erwin Lutwak上海大学，2026 年 6 月 1–5 日 · 会议主页、合影、日程与手册Shanghai University, 1–5 June 2026 · conference page, photo, program and handbook
    • 2025.11
      凸几何分析与随机几何（Convex Geometric Analysis and Stochastic Geometry）Convex Geometric Analysis and Stochastic Geometry天元数学国际交流中心，云南昆明，2025 年 11 月 24–28 日。组织者：张宁、席东盟Tianyuan Mathematics Research Center, Kunming, Yunnan, 24–28 November 2025. Organisers: Ning Zhang, Dongmeng Xi

    审稿Refereeing

      教育与工作经历Education and career

      从冶金工程到几何与分析。From metallurgical engineering to geometry and analysis.

      2008

      江苏大学Jiangsu University

      冶金工程，工学学士BEng in Metallurgical Engineering

      2012

      广西民族大学Guangxi Minzu University

      理学硕士，导师蓝师义教授MSc; advisor Prof. Shiyi Lan

      2015

      上海大学Shanghai University

      理学博士，导师冷岗松教授；同年任讲师（至 2018）PhD, advisor Prof. Gangsong Leng; lecturer the same year (to 2018)

      2016

      复旦大学Fudan University

      博士后（至 2018），合作导师傅吉祥教授Postdoctoral fellow (to 2018); mentor Prof. Jixiang Fu

      2019

      上海大学Shanghai University

      副教授（至 2023）Associate Professor (to 2023)

      2023

      上海大学Shanghai University

      教授，伟长学者Professor and Weichang Scholar

      科研项目Grants

      以下项目均为主持。All as principal investigator.

      代表性论文Selected papers

      按年份排列。Newest first.

        讲义：Dongmeng Xi, Jin Li, Lecture Notes on Functional Analysis。Lecture notes: Dongmeng Xi and Jin Li, Lecture Notes on Functional Analysis.

        联系方式Contact

        xi_dongmeng@shu.edu.cn dongmeng.xi@live.com

        上海市宝山区上大路 99 号
        上海大学理学院数学系，200444        Department of Mathematics, College of Sciences
        Shanghai University, 99 Shangda Road, Baoshan, Shanghai 200444