The Willmore functional and the containment problem in R4

Let ∑ be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional f∑ H2dσ. This bound is an invariant involving the area of ∑, the volume and Minkowski quermassintegrals of the convex body that ∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.     

ON THE WILLMORE’S THEOREM FOR CONVEX HYPERSURFACES

Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean curvature M H2dA. The bound is the Minkowski quermassintegral of the convex body K. The total square of mean curvature attains the lower bound when M is an (n-1)-sphere.     

The Willmore functional and the containment problem in R4

Let Σ be a convex hypersurface in the Euclidean space R ⁴ with mean curvature H. We obtain a geometric lower bound for the Willmore functional ∫_Σ H ² dσ. This bound is an invariant involving the area of Σ, the volume and Minkowski quermassintegrals of the convex body that Σ bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R ⁴.     

欧氏空间R3中的几个Bonnesen型不等式

本文研究欧氏空间R3中关于一个凸体的Bonnesen型不等式.利用包含测度的方法,获得了几个Bonnesen型不等式.     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.     

经典Hadamard不等式的高维推广

在n维Euclid空间利用多重积分的一般Stokes公式,将一元凸函数的经典H adam ard不等式在高维空间一般凸区域上进行了推广,得到了相应的高维Hadamard型不等式.这个结果蕴涵了经典的H adam ard不等式以及几个特殊凸体上的H adam ard型不等式.     

An Estimate for the Problem of Illumination of the Boundary of a Convex Body in E3

Let A be a convex body in Euclidean space E3. We denote by H(A) the smallest number of homothetic 'reduced copies of A by which it is possible to cover the whole of A. The conjecture of Hadwiger is H(A) 8. We prove that H(A) 16.     

Eigenvalues of the Wentzell–Laplace operator and of the fourth order Steklov problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell–Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.     

DEFORMING CONVEX HYPERSURFACES BY THE LINEAR COMBINATION OF THE MEAN CURVATURE AND THE N-TH ROOT OF THE GAUSS-KRONECKER CURVATURE

1IntroductionTherehavebeensomeinterestingresultsinstudyingtheflowofconvexhypersurfacesintheEuclideanspacebyfunctionsOftheirprincipalcurvatures.BeingviewedasanextensionOfthetheoremOfGageandHedton[3],Huiskellprovedin16]thatdeformingconvexhypersurforesbytheirmeancurysturefunctiollsconvergetoaroundsphereinasense.FollowingthemethodsOfHuiskell[6]andTso[IOI,Chowshowedin[1]thatthesame.statementasin.[6]reconstrueifthemeancurvatureisreplacedbythen-throotoftheGauss-Kroneckercurvature.FOrgenerality…     

An Improvement of an Inequality Linking Packing and Covering Densities in 3-space

It is shown that if K is a compact convex set which is centrally symmetric and has a nonempty interior, then the density of the tightest lattice packing with copies of K in Euclidean 3-space divided by the density of the thinnest lattice covering of Euclidean 3-space with copies of K is greater than or equal to 1/3. This improves the previous bound in [5] of 1/4. It is possible this bound will be improved in the future, though not beyond approximately 1/2     

非常极凸空间

本文使用非常极凸的定义，证明了非常极凸和非常光滑是互为对偶空间且严格介于弱k凸和非常凸之间的空间，最后得到了非常极凸的一些特征．     

A recursive algorithm for finding the minimum norm point in a polytope and a pair of closest points in two polytopes

For a given pair of finite point setsP andQ in some Euclidean space we consider two problems: Problem 1 of finding the minimum Euclidean norm point in the convex hull ofP and Problem 2 of finding a minimum Euclidean distance pair of points in the convex hulls ofP andQ. We propose a finite recursive algorithm for these problems. The algorithm is not based on the simplicial decomposition of convex sets and does not require to solve systems of linear equations.     

A Reilly Type Inequality for the First Eigenvalue of Finsler Submanifolds in Minkowski Space

In this paper, we study the extrinsic upper bound for the first eigenvalue of compact Finsler submanifolds in Minkowski space and obtain a Reilly type inequality. It is reduced to the standard Reilly inequality when the ambient space is Euclidean.Mathematics Subject Classification (1991): 53C60, 53B40.     

Two-point symmetrization and convexity

We prove a conjecture of R. Schneider: the spherical caps are the only spherically convex bodies of the sphere which remain spherically convex after any two-point symmetrization. More generally, we study the relationships between convexity and two-point symmetrization in the Euclidean space and on the sphere. Received: 4 April 2003     

Measure-Valued Valuations and Mixed Curvature Measures of Convex Bodies

We prove a polynomial expansion for measure-valued functionals which are translation covariant on the set of convex bodies. The coefficients are measures on product spaces. We then apply this construction to the curvature measures of convex bodies and obtain mixed curvature measures for bodies in general relative position. These are used to generalize an integral geometric formula for nonintersecting convex bodies. Finally, we introduce support measures relative to a quite general structuring body B and describe connections between the different types of measures.     

On the chromatic number for a set of metric spaces

We study the problem of finding the chromatic number of a metric space with a forbidden distance. Using the linear-algebraic technique in combinatorics and convex optimization methods, we obtain a set of new estimates and observe the change of the asymptotic lower bound for the chromatic number of Euclidean space under the continuous change of the metric from l ₁ to l ₂.     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.     

非常极凸空间的推广及其对偶概念

本文研究了k非常极凸和k非常极光滑空间的问题.利用Banach空间理论的方法,证明了k非常极凸空间和k非常极光滑空间是一对对偶概念,并且k非常极凸空间(k非常极光滑空间)是严格介于k一致极凸空间和k非常凸空间(k一致极光滑空间和k非常光滑空间)之间的一类新的Banach空间,得到了k非常极凸空间和k非常极光滑空间的若干等价刻画以及k非常极凸(k非常极光滑性)与其它凸性(光滑性)之间的蕴涵关系,推广了非常极凸空间和非常极光滑空间,完善了k非常极凸空间及其对偶空间的研究.     

曲面上的曲率新讲

阐述一种讲解曲面上的各种曲率形式的新教学模式.我们利用平面向量分析,矩阵,求导运算等工具推导出所有的结论和公式.这种新观点和方法避免了微分几何传统讲法中使用Weingarten映射及其特征值的思路,而较之更加直观明了.