关于L_p-极投影体Shephard问题的一个类似（英文）

For p 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_(-p)K of a convex body K in Rn. Let p ≥ 1 and K, L ? Rnbe two origin-symmetric convex bodies, we consider the question of whether Γ_(-p) K ? Γ_(-p) L implies ?_p(L) ≤ ?_p(K),where ?_p(K) denotes the L_p-affine surface area of K and K = Voln(K)~(-1/p) K. We prove a necessary and sufficient condition of an analog of the Shephard problem for the L_p-polar projection bodies.     

Some Inequalities for the General Radial Bo dies

Lutwak showed the radial bodies combining with the radial Minkowski linear combinations of star bodies. In this paper, the general radial bodies are introduced and its some properties and inequalities are established.     

关于L_p-相交体的Busemann-Petty型问题(英文)

Haberl and Ludwig introduced the L_p-intersection body I_pK for an originsymmetric star body K in R~n,where p < 1 and p ≠ 0.In this paper,we consider the Busemann-Petty’s problem for L_p-intersection bodies I_pK and I_pL.That is,whether I_pK ■ IpL implies Vol_n(K) ≤ Vol_n(L).We obtain that for two origin-symmetric star bodies K and L in R~n,such that(R~n,||·||K) embeds in L_p and I_pK ■ IpL,then vol_n(K) ≤ vol_n(L) for 0 < p < 1 and vol_n(K) ≥ vol_n(L) for p < 0.     

关于L_p-极曲率映像的几个不等式（英文）

Zhu,Lü and Leng extended the concept of L_p-polar curvature image. We continuously study the L_p-polar curvature image and mainly expound the relations between the volumes of star bodies and their L_p-polar curvature images in this article. We first establish the L_p-affine isoperimetric inequality associated with L_p-polar curvature image. Secondly,we give a monotonic property for L_p-polar curvature image. Finally, we obtain an interesting equation related to L_p-projection body of L_p-polar curvature image and L_p-centroid body.     

Some Inequalities for the Lp-p olar Curvature Images of Star Bo dies

Zhu, L¨u and Leng extended the concept of Lp-polar curvature image. We con-tinuously study the Lp-polar curvature image and mainly expound the relations between the volumes of star bodies and their Lp-polar curvature images in this article. We first establish the Lp-a?ne isoperimetric inequality associated with Lp-polar curvature image. Secondly, we give a monotonic property for Lp-polar curvature image. Finally, we obtain an interesting equation related to Lp-projection body of Lp-polar curvature image and Lp-centroid body.     

INEQUALITIES OF THE QUERMASSINTEGRALS FOR THE L_p-PROJECTION BODY AND THE L_p-CENTROID BODY

Associated with the concepts of the Lp-mixed quermassintegrals, the Lp-mixed volume, and the Lp-dual mixed volume, we establish inequalities for the quermassintegrals of the Lp-projection body and the Lp-centroid body. Further, the general results for the Shephard problem of the Lp-projection body and the Lp-centroid body are obtained.     

On Prop erties of p-critical Points of Convex Bo dies

Properties of the p-measures of asymmetry and the corresponding affine equivariant p-critical points, defined recently by the second author, for convex bodies are discussed in this article. In particular, the continuity of p-critical points with respect to p on(1, +∞) is confirmed, and the connections between general p-critical points and the Minkowski-critical points(∞-critical points) are investigated. The behavior of p-critical points of convex bodies approximating a convex bodies is studied as well.     

An Explicit Counter-example for the Shephard Problem of Convex Bodies in R~n

Comparing the volume of the projection body of a double cone and that of the projection body of a ball,we give an explicit counter-example for the Shephard problem of convex bodies in R~n(n≥3)and an affirmative answer to the question of Zhang.     

Some studies on mathematical models for general elastic multi-structures

The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.     

凸体间的逼近

For the affine distance d(C, D) between two convex bodies C, D C R^n, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) ≤ n^1/2 if one is an ellipsoid and another is symmetric, d(C, D) ≤ n if both are symmetric, and from F. John's result and d(C1, C2) ≤ d(C1, C3)d(C2, C3) one has d(C, D) ≤ n^2 for general convex bodies; M. Lassak proved d(C, D) ≤ (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asvmmetrv for convex bodies.     

多重星体的L_p-对偶混合几何表面积及其相关不等式

叶德平等人介绍了任意实数p(p≠-n)的多重凸体的L_p-混合几何表面积.本文给出了关于任意实数p(p≠n)的多重星体的L_p-对偶混合几何表面积的概念,并且建立了一些相关不等式.     

Lp径向Blaschke-Minkowski同态的Lp对偶几何表面积

2006年，Schuster提出了径向Blaschke-Minkowski同态的概念.随后，汪卫等人将其推广到L_p径向Blaschke-Minkowski同态.本文结合L_p对偶几何表面积，建立了L_p径向Blaschke-Minkowski同态的若干不等式，包括Brunn-Minkowski型不等式和单调不等式.并给出了L_p径向Blaschke-Minkowski同态的Busemann-Petty问题的肯定和否定形式.     

General L p -harmonic Blaschke bodies

Lutwak introduced the harmonic Blaschke combination and the harmonic Blaschke body of a star body. Further, Feng and Wang introduced the concept of the L _p -harmonic Blaschke body of a star body. In this paper, we define the notion of general L _p -harmonic Blaschke bodies and establish some of its properties. In particular, we obtain the extreme values concerning the volume and the L _p -dual geominimal surface area of this new notion.     

Upper bounds for the perimeter of plane convex bodies

We show that the maximum total perimeter of k plane convex bodies with disjoint interiors lying inside a given convex body C is equal to $\operatorname{per}\, (C)+2(k-1)\operatorname{diam}\, (C)$ , in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a bound on the perimeter of a polygon with at most k reflex angles lying inside a given plane convex body.     

The geometry of p-convex intersection bodies

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.     

Sectional bodies associated with a convex body

We define the sectional bodies associated to a convex body in and two related measures of symmetry. These definitions extend those of Grünbaum (1963). As Grünbaum conjectured, we prove that the simplices are the most dissymmetrical convex bodies with respect to these measures. In the case when the convex body has a sufficiently smooth boundary, we investigate some limit behaviours of the volume of the sectional bodies.

     

(p,q)-John椭球

本文主要研究(p,q)-John椭球.经典的John椭球和L_p John椭球均是(p,q)-John椭球的特殊情形.首先讨论(p,q)-John椭球的充分必要条件和连续性.得到了关于(p,q)-John椭球的不等式和包含关系,所得到的不等式和包含关系分别类似于Ball体积比不等式和John包含关系.     

The pyramidal configurations of N+1 bodies cannot rotate

For the (N+1)-body problem, we assume that N bodies are at the vertices of a unit regular polygon and the (N+1)st body is along the vertical line normal to the plane formed by the former N bodies. If N bodies rotate at the unit circle and the (N+1)st body oscillates along the vertical line of the plane formed by the former N bodies and passing through the geometrical center, then we prove that the (N+1)st body must locate at the geometrical center of unit regular polygon.