THE UNIQUENESS OF THE L_(p) MINKOWSKI PROBLEM FOR q-TORSIONAL RIGIDITY

In this paper,we prove the uniqueness of the Lp Minkowski problem for q-torsional rigidity with p>1 and q>1 in smooth case.Meanwhile,the Lp Brunn-Minkowski inequality and the Lp Hadamard variational formula for q-torsional rigidity are established.

On Existence of the Even L_(p)Gaussian Minkowski Problem for p>n

This paper concerns the even L_(p)Gaussian Minkowski problem in n-dimensional Euclidean space R^(n).The existence of the solution to the even L_(p)Guassian Minkowski problem for p>n is obtained.

The UNIFORM C^(0)ESTIMATE AND WEIGHTED ESTIMATE OF GENERALIZED CHRISTOFFEL-MINKOWSKI PROBLEMS

In this paper,we consider generalized Christo®el-Minkowski problems as followsσ_(k)(u_(ij)+uδ_(ij))/σ_(l)(u_(ij)+uδ_(ij))=|u^(p-1)f(x),x∈S^(n),where 0≤l≤k≤n,p-1>0 and f is positive,and we establish the weighted gradient estimate and uniform C^(0)estimate for the positive convex even solutions,which is a generalization of Guan-Xia[1]and Guan[2].

与欧氏空间共形的背景空间中的Minkowski问题

作者研究了与欧氏空间共形的背景空间中的Minkowski问题,并利用连续性方法证明了该背景空间下Minkowski问题的可解性.

The Logarithmic Capacitary Minkowski Problem for Polytopes

The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body.This article completely solves the case of discrete measures whose support sets are in general position.

Continuity of the Solution to the L_(p) Minkowski Problem in Gaussian Probability Space

In this paper,it is proved that the weak convergence of the L_(p) Gaussian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p≥1.Moreover,continuity of the solution to the L_(p) Gaussian Minkowski problem with respect to p is obtained.

The planar L_(p) dual Minkowski problem

In this paper we study the L_(p) dual Minkowski problem for the case p<0<q.We prove for any positive smooth function f on S^(1),there exists an F:R^(+)→R^(-),such that if F(q)<p<0 or 0<q<-F(-p)then there is a smooth and strictly convex body solving the planar L_(p) dual Minkowski problem.

Nonexistence of maximizers for the functional of the centroaffine Minkowski problem

The centroafhine Minkowski problem is studied, which is the critical case of the L_p-Minkowski problem. It admits a variational structure that plays an important role in studying the existence of solutions.In this paper, we find that there is generally no maximizer of the corresponding functional for the centroaffine Minkowski problem.

Continuity of the solution to the even logarithmic Minkowski problem in the plane

This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in R^2 implies the convergence of the sequence of the corresponding origin-symmetric convex bodies in the Hausdorff metric.

L_(p)-Minkowski问题周期解的存在性

L_(p)-Minkowski问题是凸几何分析L_(p)-Brunn-Minkowski理论的核心,其实质是分析给定测度是否为凸体的L_(p)表面积测度问题.这个问题可以简化为二阶微分方程的周期解的存在性,L_(p)-Minkowski问题中周期解的存在性问题如下u″+u=h(t)/u^(ρ),其中h>0是连续的周期函数,常数ρ=1-p.利用了二阶微分方程周期解存在的充分条件,通过建立的一个方程的周期解的存在性判据,再利用Sobolev inequality证明了这个二阶微分方程周期解的存在性,得到在一定条件下周期解存在,这个方法在一定程度上扩大p的取值范围.最后给出一个例子,验证文中所得到的主要结果的可行性.

2维不规则条带装箱问题的一种启发式求解算法

不规则条带装箱问题是切分与布局问题的子问题,且在工业制造领域中尤为常见.该文针对2维不规则单箱装箱问题(two-dimensional irregular single bin size bin packing problem)中的特殊变种条带装箱问题,提出结合闵可夫斯基和(即闵可夫斯基加法)与遗传算法的综合求解的算法,其中遗传算法在优化算法中起到整体框架的作用,并在通过闵可夫斯基和寻找到候选放置位置后使用启发式的方法进行放置,同时讨论了在特定情况下该算法与传统优化方法的对比效果.在实际数据测试(50块数据集)中,较成熟的商业软件SVGnest得到的最终面积利用率为78.94%,应用该文的启发式算法,最终面积实用率提升到81.2%.

Grushin平面上的Brunn-Minkowski不等式和等周不等式

首先证明了Grushin球不是Grushin平面上等周问题的解,然后得到了Brunn-Minkowski不等式在Grushin平面上是不成立的。

R^2上对偶Minkowski问题的可解性

该文研究Sobolev空间W^1,4(S)中一类约束变分问题存在极小可达元,在∫Sg(θ)dθ>0条件下,该极小可达元是相应Euler-Lagrange方程u″+u=g(θ)/u(u^2+u′^2),θ∈S的严格正解.基于此,该文在R^2上证明了文献[Huang-Lutwak-Yang-Zhang.Acta Math,2016,216(2):325-338]提出的对偶Minkowski问题的可解性.

Hyperbolic Yamabe problem

In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the （1 ＋ n）-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n =2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem; while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.

L_p-Minkowski问题椭球解的唯一性（英文）

本文研究了L_p-Minkowski问题(解是中心在原点的椭球的假定下).利用支撑函数与高斯曲率的关系,获得了当p<1时椭球解的唯一性,推广了L_p-Minkowski问题以及L_p-和的Christoffel-Minkowski问题的唯一性结果.

零点到两个闭半代数集的Minkowski和上的投影问题的数值算法

Minkowski和是计算几何的一个重要概念,在路径规划、动态仿真等领域有着重要作用.提出一种计算零点到两个闭半代数集的Minkowski和上投影的数值算法.首先将问题转化为多项式优化问题,然后采用Lasserre半正定松弛方法对该问题进行求解,接着给出相应的数值实验,数值实验的结果验证了所提算法的有效性.

Lp-Blaschke-Minkowski同态的Shephard问题

Schuster定义了Blaschke-Minkowski同态的概念之后,汪卫又将此概念推广到了Lp形式.在本文中,结合Lp仿射表面积,主要研究了Lp-Blaschke-Minkowski同态的Shephard问题.

Applications of new affine invariant for polytopes

To study the Schneider＇s projection problem, Lutwak, Yang and Zhang recently introduced a new .affine invariant functional U（P） for convex polytopes in R^n. In the paper, we obtain the analytic expression of the affine-invariant U（P） defined on a specific subclass of origin-symmetric convex polytopes in Rn and give an application of U（P） to the Lp-Minkowski problem.

广义共同逼近问题的适定性

设 C是实 Banach空间 X中有界闭凸子集且 0是 C的内点 ,G是 X中非空闭的有界相对弱紧子集 .记 K( X)为 X的非空紧凸子集全体并赋 H ausdorff距离 ,KG( X)为集合 {A∈ K( X) ;A∩ G=}的闭包 .称广义共同逼近问题 min C( A,G)是适定的是指它有唯一解 ( x0 ,z0 ) ,且它的每个极小化序列均强收敛到 ( x0 ,z0 ) .在 C是严格凸和 Kadec的假定下 ,证明了 {A∈K( X) ;min C( A,G)是适定的 }含有 KG( X)中稠 Gδ子集 ,这本质地推广和延拓了包括 De Blasi,Myjak andPapini[1]、Li[2 ]和 De Blasi and Myjak[3]等人在内的近期相应结果 .

A Type of Busemann-Petty Problems for Blaschke-Minkowski Homomorphisms

Schuster introduced the notion of Blaschke-Minkowski homomorphisms and first considered Busemann-Petty type problems. In this paper, we study the Busemann-Petty type problems for the geominimal surface area with respect to Blaschke-Minkowski homomorphisms.